nsc.ru · ÓÄÊ 517 ÁÁÊ B16 Ñ545 Ñîáîëåâñêèå ÷òåíèÿ....

SOBOLEV READINGS

International School-ConferenceNovosibirsk, Russia, August 20-23, 2017

ABSTRACTS

СОБОЛЕВСКИЕ ЧТЕНИЯ

Международная Школа-КонференцияНовосибирск, Россия, 20-23 августа 2017

ТЕЗИСЫ ДОКЛАДОВ

ÐÎÑÑÈÉÑÊÀß ÀÊÀÄÅÌÈß ÍÀÓÊÑÈÁÈÐÑÊÎÅ ÎÒÄÅËÅÍÈÅ

ÈÍÑÒÈÒÓÒ ÌÀÒÅÌÀÒÈÊÈ èì. Ñ.Ë. ÑÎÁÎËÅÂÀ

ÌÈÍÈÑÒÅÐÑÒÂÎ ÎÁÐÀÇÎÂÀÍÈß È ÍÀÓÊÈ ÐÔÍÎÂÎÑÈÁÈÐÑÊÈÉ ÃÎÑÓÄÀÐÑÒÂÅÍÍÛÉ

ÓÍÈÂÅÐÑÈÒÅÒ

ÑÎÁÎËÅÂÑÊÈÅ ×ÒÅÍÈß

Ìåæäóíàðîäíàÿ øêîëà-êîíôåðåíöèÿ

Íîâîñèáèðñê, Ðîññèÿ, 2023 àâãóñòà 2017 ã.

ÒÅÇÈÑÛ ÄÎÊËÀÄÎÂ

ÍÎÂÎÑÈÁÈÐÑÊ

2017

ÓÄÊ 517ÁÁÊ B16

Ñ545

Ñîáîëåâñêèå ÷òåíèÿ. Ìåæäóíàðîäíàÿ øêîëà-êîíôåðåíöèÿ (Íîâîñè-áèðñê, 2023 àâãóñòà 2017 ã.): Òåç. äîêëàäîâ / ïîä ðåä. Ã. Â. Äåìèäåíêî. Íîâîñèáèðñê: Èçä-âî Èíñòèòóòà ìàòåìàòèêè, 2017. 135 ñ.

ISBN 978-5-86134-208-7

Îðãàíèçàòîðû

Èíñòèòóò ìàòåìàòèêè èì. Ñ.Ë. Ñîáîëåâà ÑÎ ÐÀÍ

Íîâîñèáèðñêèé ãîñóäàðñòâåííûé óíèâåðñèòåò

Îòâåòñòâåííûé ðåäàêòîð: Ã.Â. Äåìèäåíêî

Organizers

Sobolev Institute of Mathematics SB RAS

Novosibirsk State University

Editor-in-Chief: G.V. Demidenko

Ñ1602070100− 03

ß82(03)− 2017Áåç îáúÿâë.

ISBN 978-5-86134-208-7

c⃝ Èíñòèòóò ìàòåìàòèêèèì. Ñ.Ë. Ñîáîëåâà ÑÎ ÐÀÍ, 2017

Ïðîãðàììíûé êîìèòåò

Ã.Â. Äåìèäåíêî ïðåäñåäàòåëü, È.È. Ìàòâååâà ñåêðåòàðü,Â.Ñ. Áåëîíîñîâ, Î.Â. Áåñîâ, À.Ì. Áëîõèí, Â.Ë. Âàñêåâè÷,Ñ.Ê. Ãîäóíîâ, Ï.È. Ïëîòíèêîâ, Þ.Ã. Ðåøåòíÿê, Â. Ã. Ðîìàíîâ,Â.Ä. Ñòåïàíîâ, À.À. Òîëñòîíîãîâ, H. Begehr, E. Feireisl, L. Hatvani,A. Laptev, R. McOwen

Îðãàíèçàöèîííûé êîìèòåò

Ã.Â. Äåìèäåíêî ñîïðåäñåäàòåëü, Ì.Ï. Ôåäîðóê ñîïðåäñåäàòåëü,Ë.Í. Áîíäàðü ñåêðåòàðü, Å.Þ. Áàëàêèíà, Ì.À. Ñêâîðöîâà,È.À. Óâàðîâà, Ò.Ê. Ûñêàê, Â.Ý. Ýéñíåð

Program Committee

G.V. Demidenko (Chairman), I. I. Matveeva (Secretary), H. Begehr,V. S. Belonosov, O.V. Besov, A.M. Blokhin, E. Feireisl, S.K. Godunov,L. Hatvani, A. Laptev, R. McOwen, P. I. Plotnikov, Yu.G. Reshetnyak,V.G. Romanov, V.D. Stepanov, A.A. Tolstonogov, V. L. Vaskevich

Organizing Committee

G.V. Demidenko (Co-Chairman), M.P. Fedoruk (Co-Chairman),L.N. Bondar (Secretary), E.Yu. Balakina, V. E. Eisner,M.A. Skvortsova, I. A. Uvarova, T.K. Yskak

Øêîëó-êîíôåðåíöèþ ïîääåðæàëè:

Ðîññèéñêèé ôîíä ôóíäàìåíòàëüíûõ èññëåäîâàíèé(ïðîåêò 17-31-10222)

Ñèáèðñêîå îòäåëåíèå Ðîññèéñêîé àêàäåìèè íàóê

Ìýðèÿ ãîðîäà Íîâîñèáèðñêà

Àâèàêîìïàíèÿ Àýðîôëîò

Ìåõàíèêî-ìàòåìàòè÷åñêèé ôàêóëüòåòÍîâîñèáèðñêîãî ãîñóäàðñòâåííîãî óíèâåðñèòåòà

The School-Conference is supported by:

Russian Foundation for Basic Research(project no. 17-31-10222)

Siberian Branch of the Russian Academy of Sciences

The Mayor's Oce of Novosibirsk

Airline Aeroot

Department of Mechanics and Mathematics,Novosibirsk State University

ÑÎÄÅÐÆÀÍÈÅ

ÖÈÊËÛ ËÅÊÖÈÉ

LECTURE COURSES

Ãðèíåñ Â. Ç.Î òîïîëîãè÷åñêîé êëàññèôèêàöèè äèíàìè÷åñêèõ ñèñòåì íà ìíîãî-îáðàçèÿõ ðàçìåðíîñòè 2 è 3On topological classication of dynamical systems on manifolds ofdimensions 2 and 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Çâÿãèí Â. Ã.Îá îäíîì íîâîì ìåòîäå èññëåäîâàíèÿ ñëàáîé ðàçðåøèìîñòè çàäà÷ãèäðîäèíàìèêèOn a new approach to investigation of weak solvability for hydro-dynamics problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

Feireisl E.Mathematics of uids in motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

×å÷êèíà À. Ã.Î ïåðâîé íàó÷íîé ðàáîòå Ñ.Ë. ÑîáîëåâàOn the rst S. L. Sobolev's scientic work . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

ÑÅÊÖÈÎÍÍÛÅ ÄÎÊËÀÄÛ

SHORT COMMUNICATIONS

Àáàøååâà Í.Ë.Ëèíåéíàÿ îáðàòíàÿ çàäà÷à äëÿ îïåðàòîðíî-äèôôåðåíöèàëüíîãîóðàâíåíèÿ ñìåøàííîãî òèïà ñ ïàðàìåòðîìLinear inverse problem for mixed type operator-dierential equationwith a parameter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Àëåêñàíäðîâ Â.Ì.Îáùåå ðåøåíèå çàäà÷è ìèíèìèçàöèè ðàñõîäà ðåñóðñàThe general solution to the problem of minimizing resourceconsumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

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Àíèêîíîâ Ä.Ñ.Ìåòîä ïëîñêèõ ñðåäíèõ äëÿ ãèïåðáîëè÷åñêèõ ñèñòåìThe plane means method for hyperbolic systems . . . . . . . . . . . . . . . . . . . . 31

Àíèêîíîâ Þ.Å., Àþïîâà Í.Á., Íåùàäèì Ì.Â.Ëó÷åâîé ìåòîä è âîïðîñû èäåíòèôèêàöèè óðàâíåíèé òåîðèè óïðó-ãîñòèRay method and questions of identication of the elasticity theoryequations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Áàëàêèíà Å.Þ.Îïðåäåëåíèå ïîâåðõíîñòåé ðàçðûâîâ êîýôôèöèåíòîâ óðàâíåíèÿïåðåíîñàFinding of surfaces of discontinuities of coecients of transfer equation 33

Áàëäàíîâ Ä.Ø.Óñëîâèÿ àñèìïòîòè÷åñêîé óñòîé÷èâîñòè ðåøåíèé ðàçíîñòíûõ óðàâ-íåíèé ñ ïåðèîäè÷åñêèìè êîýôôèöèåíòàìè ñ çàïàçäûâàíèåìConditions for the asymptotic stability of solutions to dierenceequations with periodic coecients and with delay . . . . . . . . . . . . . . . . . . 34

Áåëîçóá Â.À., Êîçëîâà Ì. Ã., Ëóêüÿíåíêî Â.À.Óðàâíåíèÿ òèïà Óðûñîíà â çàäà÷àõ âîññòàíîâëåíèÿ òî÷åê ïîâåðõ-íîñòèUrysohn type equations in problems of recovering surface points . . . . . 35

Áèáåðäîðô Ý.À., Áëîõèí À.Ì.Êðàåâàÿ çàäà÷à îá îñåñèììåòðè÷íîì îáòåêàíèè êðóãîâîãî êîíóñàðåàëüíûì ãàçîìThe boundary value problem on the axisymmetric ow of a real gasabout a circular cone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

Áîëäûðåâ À.Ñ.Òðàåêòîðíûé è ãëîáàëüíûé àòòðàêòîðû îäíîãî êëàññà âÿçêîóïðó-ãèõ æèäêîñòåé ñ ïàìÿòüþTrajectory and global attractors of one class of viscoelastic uids withmemory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

Áîíäàðü Ë.Í.O ðàçðåøèìîñòè âòîðîé êðàåâîé çàäà÷è äëÿ ñèñòåìû ÑòîêñàOn solvability of the second boundary value problem for the Stokessystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

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Áîðîâñêèõ À.Â., Ïëàòîíîâà K.C.Ãðóïïîâîé àíàëèç îäíîìåðíîãî óðàâíåíèÿ ÁîëüöìàíàGroup analysis of the one-dimensional Boltzmann equation . . . . . . . . . . 40

Âàñêåâè÷ Â.Ë.Ñõîäèìîñòü êóáàòóðíûõ ôîðìóë âûñîêîé òðèãîíîìåòðè÷åñêîéòî÷íîñòèConvergence of cubature formulas of high trigonometric degree . . . . . . 41

Âîëîêèòèí Å.Ï.Àëãåáðàè÷åñêèå ïåðâûå èíòåãðàëû ïîëèíîìèàëüíûõ ñèñòåì, óäî-âëåòâîðÿþùèõ óñëîâèÿì Êîøè ÐèìàíàAlgebraic rst integrals of the polynomial systems satisfying theCauchyRiemann conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

Ãðåøíîâ À.Â.Î òîïîëîãèè (q1, q2)-êâàçèìåòðè÷åñêèõ ïðîñòðàíñòâOn the topology of (q1, q2)-quasimetric spaces . . . . . . . . . . . . . . . . . . . . . . . 43

Äâîðíèöêèé Â.ß.Îá óñòîé÷èâîñòè ðåøåíèé îäíîãî êëàññà ñèñòåì íåéòðàëüíîãî òèïàñ ïàðàìåòðîìOn stability of solutions to one class of neutral-type systems with aparameter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

Äåìèäåíêî Ã.Â.Î çàäà÷å Êîøè äëÿ ïñåâäîãèïåðáîëè÷åñêèõ óðàâíåíèéOn the Cauchy problem for pseudo-hyperbolic equations . . . . . . . . . . . . 45

Äåíèñîâà Ò. Å.Îá îäíîì ìåòîäå èçó÷åíèÿ êà÷åñòâåííûõ ñâîéñòâ ðåøåíèé ïåðâîéíà÷àëüíî-êðàåâîé çàäà÷è äëÿ óðàâíåíèé ñîáîëåâñêîãî òèïàOn a method of the study of qualitative properties of solutions to therst initial-boundary value problem for Sobolev-type equations . . . . . . 46

Äæåíàëèåâ Ì.Ò., Èñêàêîâ Ñ.À., Ðàìàçàíîâ Ì.È.Ê òåîðèè ãðàíè÷íîé çàäà÷è Ñîëîííèêîâà ÔàçàíîOn the theory of the SolonnikovFasano boundary value problem. . . . 48

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Åãîðîâ À.À.Ñâîéñòâà ã¼ëüäåðîâîé íåïðåðûâíîñòè ðåøåíèé äèôôåðåíöèàëü-íûõ íåðàâåíñòâ ñ íåêîòîðûìè íóëü-ëàãðàíæèàíàìèThe Holder continuity properties of solutions to dierential inequalitieswith some null Lagrangians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

Åãîðøèí À.Î.Îöåíèâàíèå êîýôôèöèåíòîâ àâòîíîìíûõ ëèíåéíûõ óðàâíåíèéCoecients estimation for autonomous linear equations . . . . . . . . . . . . . 50

Åëåóîâ À.À., Çàêàðèÿíîâà Í.Á., Åëåóîâà Ð.À.Î åäèíñòâåííîñòè ðåøåíèÿ îáðàòíîé çàäà÷è ñïåêòðàëüíîãî àíà-ëèçà äëÿ äèôôåðåíöèàëüíûõ îïåðàòîðîâ âûñøèõ ïîðÿäêîâ íà îò-ðåçêåOn the uniqueness of a solution to the inverse problem of spectralanalysis for the higher-order dierential operators on an interval . . . . . 51

Çâÿãèí À.Â.Çàäà÷à îïòèìàëüíîãî óïðàâëåíèÿ ñ îáðàòíîé ñâÿçüþ äëÿ òåðìî-âÿçêîóïðóãîé ìîäåëè Êåëüâèíà ÔîéãòàThe optimal control problem with feedback for the thermoviscoelasticKelvinVoigt model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

Çâÿãèí Â. Ã., Òóðáèí Ì.Â.Àòòðàêòîðû ìîäåëè Áèíãàìà ñ ïåðèîäè÷åñêèìè óñëîâèÿìè ïî ïðî-ñòðàíñòâåííûì ïåðåìåííûì â òð¼õìåðíîì ñëó÷àåAttractors for Bingham model with periodic boundary conditions inthree-dimensional case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

Çèêèðîâ Î.Ñ.Îá îäíîé çàäà÷å äëÿ óðàâíåíèÿ òðåòüåãî ïîðÿäêà ñîñòàâíîãî òèïàOn a problem for the third-order equation of composite type . . . . . . . . 55

Çîëîòîòðóáîâà Ã.Î.Èññëåäîâàíèå íà÷àëüíî-êðàåâîé çàäà÷è äëÿ ñèñòåìû óðàâíåíèéÍàâüå Ñòîêñà ñ ïåðåìåííîé ïëîòíîñòüþInvestigation of the initial-boundary value problem for the NavierStokes system with variable density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

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Èáðàãèìîâà Í.À.Èíòåãðàëüíîå ïðåäñòàâëåíèå ðåøåíèÿ îäíîãî âûðîæäàþùåãîñÿÂ-ýëëèïòè÷åñêîãî óðàâíåíèÿ ñ ïîëîæèòåëüíûì ïàðàìåòðîìIntegral representation of a solution to a degenerating B-elliptic equa-tion with positive parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

Èâàíîâ Â.Â.Èíòåãðèðóþùèé ìíîæèòåëü ïàðû íåçàâèñèìûõ ôîðì ÏôàôôàAn integrating multiplier of a pair of independent Pfaan forms . . . . 59

Êàáàíèõèí Ñ.È., Êðèâîðîòüêî Î.È., Êîíäàêîâà Å.À.Ìåòîä îïòèìàëüíîãî óïðàâëåíèÿ â çàäà÷å äëÿ ìàòåìàòè÷åñêîéìîäåëè âçàèìîäåéñòâèÿ êëåòîê íîâîîáðàçîâàíèÿ è èììóíèòåòà âóñëîâèÿõ ðàäèîòåðàïèèThe optimal control method in a problem for mathematical model ofcancer treatment by radiotherapy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

Êàáàíèõèí Ñ.È., Øèøëåíèí Ì.À., Øîëïàíáàåâ Á.Á.×èñëåííîå ðåøåíèå íà÷àëüíî-êðàåâîé çàäà÷è äëÿ óðàâíåíèÿ ýëåê-òðîäèíàìèêèNumerical solving the initial-boundary value problem for the equationof electrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

Êîæàíîâ À.È.Êðàåâûå çàäà÷è äëÿ íåêîòîðûõ êëàññîâ ñèëüíî íåëèíåéíûõ óðàâ-íåíèé ñîáîëåâñêîãî òèïàBoundary value problems for some classes of strongly nonlinearSobolev type equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

Êîíîâàëîâà Ä.Ñ.Îäíîìåðíîå âîëíîâîå óðàâíåíèå äëÿ íåîäíîðîäíîé ñðåäûOne-dimensional wave equation for inhomogeneous medium . . . . . . . . . 65

Êîíîíîâ À.Ä.Î ðîáàñòíîé óñòîé÷èâîñòè ñòàöèîíàðíûõ äèôôåðåíöèàëüíî-àëãåá-ðàè÷åñêèõ óðàâíåíèé ñî ñòðóêòóðèðîâàííîé íåîïðåäåëåííîñòüþOn robust stability of stationary dierential-algebraic equations withstructured uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

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Êóçíåöîâ Ï.À.Ê âîïðîñó îá àíàëèòè÷åñêîé ðàçðåøèìîñòè çàäà÷ ñ âûðîæäåíèåìäëÿ íåëèíåéíîãî óðàâíåíèÿ òåïëîïðîâîäíîñòèOn the question of analytic solvability of problems with degenerationfor the nonlinear heat equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

Ëàòûïîâ È.È.Àñèìïòîòèêà ðåøåíèÿ íåñòàöèîíàðíîé ñèíãóëÿðíî-âîçìóùåííîéêðàåâîé çàäà÷è òåïëîìàññîîáìåíàAsymptotics of a solution to the nonstationary singularly perturbedboundary value problem of heat and mass transfer . . . . . . . . . . . . . . . . . . 69

Ëàòûøåíêî Â.À., Êðèâîðîòüêî Î.È., Êàáàíèõèí Ñ.È.Ìåòîäû ïðàêòè÷åñêîé èäåíòèôèöèðóåìîñòè íåêîòîðûõ ìàòåìàòè-÷åñêèõ ìîäåëåé áèîëîãèèThe practical identiability methods for some mathematical models ofbiology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

Ëàøèíà Å.À., ×óìàêîâà Í.À., ×óìàêîâ Ã.À.Îá óñòîé÷èâîñòè íà êîíå÷íîì èíòåðâàëå âðåìåíè óòîê-öèêëîâ îä-íîé êèíåòè÷åñêîé ìîäåëèOn the stability of canard-cycles of one kinetic model on a nite timeinterval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

Ëîìàêèí À.Â.Óñëîâèÿ ñòàáèëèçàöèè ðåøåíèé êðàåâûõ çàäà÷ äëÿ óðàâíåíèÿ Ñî-áîëåâàConditions for stabilization of solutions to boundary value problemsfor the Sobolev equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

Ëîìàêèíà Å.À.Îöåíêè ðåøåíèé âòîðîé êðàåâîé çàäà÷è äëÿ óðàâíåíèÿ ÑîáîëåâàEstimates of solutions to the second boundary value problem for theSobolev equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

Ëîìîâ À.À.Èññëåäîâàíèå ñâîéñòâ îöåíîê ïàðàìåòðîâ ðàçíîñòíûõ óðàâíåíèéíà êîíå÷íûõ âûáîðêàõ ëèíåàðèçàöèåé ãðàäèåíòà öåëåâûõ ôóíê-öèéInvestigation of properties of the parameters estimates for dierenceequations on nite samples by linearizing the gradient of targetfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

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Ëîìîâ À.À., Ôåäîñååâ À.Â.Îöåíêè ÷óâñòâèòåëüíîñòè èäåíòèôèêàöèè êîýôôèöèåíòîâ ðàç-íîñòíûõ óðàâíåíèé ÷åðåç ëîêàëüíûå ðàçëîæåíèÿ öåëåâûõ ôóíê-öèéEstimates of sensitivity of identication of dierence equationscoecients using local decompositions of target functions . . . . . . . . . . . . 75

Ëóêèíà Ã.À.Íåëîêàëüíûå êðàåâûå çàäà÷è ñ ÷àñòè÷íî èíòåãðàëüíûìè óñëîâèÿ-ìè äëÿ âûðîæäàþùèõñÿ äèôôåðåíöèàëüíûõ óðàâíåíèéNonlocal boundary value problems with partially integral conditionsfor degenerate dierential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

Ìàëþòèíà Ì.Â., Îðëîâ Ñ.Ñ.Ïåðèîäè÷åñêîå ðåøåíèå èíòåãðàëüíîãî óðàâíåíèÿ Àáåëÿ ïåðâîãîðîäàPeriodic solution to Abel integral equation of the rst kind . . . . . . . . . . 77

Ìàìîíòîâ À. Å., Ïðîêóäèí Ä.À.Ãëîáàëüíàÿ ðàçðåøèìîñòü íà÷àëüíî-êðàåâîé çàäà÷è äëÿ îäíîìåð-íûõ óðàâíåíèé äèíàìèêè ìíîãîñêîðîñòíûõ ñìåñåéGlobal solvability of the initial-boundary value problem for one-dimensional equations of dynamics of multi-speed mixtures . . . . . . . . . . 78

Ìàòâååâà È.È.Ðîáàñòíàÿ óñòîé÷èâîñòü ðåøåíèé ëèíåéíûõ ïåðèîäè÷åñêèõ ñèñòåìíåéòðàëüíîãî òèïà ñ íåñêîëüêèìè çàïàçäûâàíèÿìèRobust stability of solutions to linear periodic systems of neutral typewith several delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

Ìèðåíêîâ Â. Å.Íåêîððåêòíûå çàäà÷è ìåõàíèêèIll-posed problems of mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

Íèêèòåíêî Å.Â.Àñèìïòîòè÷åñêèå ñâîéñòâà ðåøåíèé íåîäíîðîäíîãî óðàâíåíèÿâíóòðåííèõ âîëíAsymptotic properties of solutions to the inhomogeneous equation ofinternal waves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

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Îðëîâ Ñ.Ñ., Øåìåòîâà Â.Â.Îáîáùåííîå ðåøåíèå äèôôåðåíöèàëüíî-îïåðàòîðíîãî óðàâíåíèÿ ñîòêëîíÿþùèìñÿ àðãóìåíòîìGeneralized solution to a dierential-operator equation with deviatingargument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

Ïèìàíîâ Ä.Î., Ôàäååâ Ñ.È., Êîñöîâ Ý. Ã.Èññëåäîâàíèå ìàòåìàòè÷åñêèõ ìîäåëåé ìèêðîýëåêòðîìåõàíè÷å-ñêîãî ðåçîíàòîðà òèïà ïëàòôîðìàInvestigation of mathematical models of microelectromechanicalplatform type resonator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

Ïîñòíîâ Ñ.Ñ.Çàäà÷è îïòèìàëüíîãî óïðàâëåíèÿ äëÿ ñèñòåì äðîáíîãî ïîðÿäêà ñðàñïðåäåë¼ííûìè ïàðàìåòðàìèOptimal control problems for systems of fractional order withdistributed parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

Ðàìàçàíîâ Ì.Ä.Òî÷íûå òåîðåìû î ïðîäîëæåíèÿõ ñ ìèíèìàëüíîé íîðìîéExact extension theorems with minimal norm . . . . . . . . . . . . . . . . . . . . . . . 88

Ñàáèòîâ Ê.Á., Ñèäîðîâ Ñ.Í.Îáðàòíûå çàäà÷è ïî îïðåäåëåíèþ ïðàâûõ ÷àñòåé óðàâíåíèé ñìå-øàííîãî òèïàInverse problems of determining the right-hand sides of mixed typeequations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

Ñåäîâà Í.Î.Îá îöåíêàõ ðåøåíèé íåàâòîíîìíîãî ëèíåéíîãî óðàâíåíèÿ ñ çàïàç-äûâàíèåìOn estimates of solutions to a time-varying linear delay equation . . . . 91

Ñåðîâàéñêèé Ñ.ß., Íóðñåèòîâ Ä.Á., Àçèìîâ À.À.Ìàòåìàòè÷åñêîå ìîäåëèðîâàíèå ãîðìîíàëüíîãî ëå÷åíèÿ îíêîëîãè-÷åñêèõ çàáîëåâàíèé â óñëîâèÿõ ãîðìîíîðåçèñòåíòíîñòèMathematical modeling of hormonal treatment of oncological diseasesunder hormonal resistance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

Ñêâîðöîâà Ì.À.Óñòîé÷èâîñòü ïîëîæåíèé ðàâíîâåñèÿ â îäíîé ìîäåëè çàáîëåâàíèÿStability of equilibrium points in a model of decease . . . . . . . . . . . . . . . . . 95

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Òàëûøåâ À.À.Äèôôåðåíöèàëüíî-èíâàðèàíòíûå ðåøåíèÿ è êðàòíûå âîëíûDierential-invariant solutions and multiple waves . . . . . . . . . . . . . . . . . . . 96

Òåëåøåâà Ë.À.Ëèíåéíàÿ îáðàòíàÿ çàäà÷à ñ ãðàíè÷íûì ïåðåîïðåäåëåíèåì â ìíî-ãîìåðíîì ñëó÷àåLinear inverse problem with boundary overdetermination in multi-dimensional case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

Óâàðîâà È.À.Àñèìïòîòè÷åñêîå ïîâåäåíèå ðåøåíèé îäíîãî óðàâíåíèÿ ñ çàïàçäû-âàþùèì àðãóìåíòîìAsymptotic behavior of solutions to a delay dierential equation. . . . . 98

Óñòþæàíèíîâà À.Ñ., Òóðáèí Ì.Â.Ðàçðåøèìîñòü â ñëàáîì ñìûñëå íà÷àëüíî-êðàåâîé çàäà÷è äëÿ ñè-ñòåìû Îñêîëêîâà ÏàâëîâñêîãîSolvability in the weak sense of the initial-boundary value problem forOskolkovPavlovsky system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

Ôåäîðîâ Â. Å.Ïðèìåíåíèå ñòàöèîíàðíîãî ìåòîäà Ãàëåðêèíà ê îäíîé êðàåâîé çà-äà÷å äëÿ óðàâíåíèÿ âûñîêîãî ïîðÿäêà ñ ìåíÿþùèìñÿ íàïðàâëåíè-åì âðåìåíèApplication of stationary Galerkin method to one boundary valueproblem for the higher-order equation with alternating time direction 100

Õàçîâà Þ.À.Ðåøåíèÿ ïàðàáîëè÷åñêîé çàäà÷è ñ ïðåîáðàçîâàíèåì ïîâîðîòà íàîêðóæíîñòèSolutions to a parabolic problem with rotation on a circle . . . . . . . . . . . 102

×àíûøåâ À.È., Àáäóëèí È.Ì., Áåëîóñîâà Î. Å.Ðåøåíèÿ çàäà÷ Êîøè â ãåîìåõàíèêåSolving the Cauchy problems in geomechanics . . . . . . . . . . . . . . . . . . . . . . . 103

×óåøåâà Í.À.Óðàâíåíèå ÊäÔ ïÿòîãî ïîðÿäêàThe KdV equation of the fth order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

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Øàäðèíà Í.Í.Èññëåäîâàíèå ðàçðåøèìîñòè êðàåâûõ çàäà÷ ñ äîïîëíèòåëüíûìèóñëîâèÿìè ñîïðÿæåíèÿInvestigation of the solvability of boundary value problems withadditional matching conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

Øàìàíàåâ Ï.À., ßçîâöåâà Î.Ñ.Ëîêàëüíàÿ àñèìïòîòè÷åñêàÿ ýêâèâàëåíòíîñòü è åå ïðèìåíåíèå êèññëåäîâàíèþ óñòîé÷èâîñòè ïî ÷àñòè ïåðåìåííûõLocal asymptotic equivalence and its application to the study ofstability with respect to a part of variables . . . . . . . . . . . . . . . . . . . . . . . . . . 106

Øèøêàíîâà À.À.Âíåöåíòðåííîå íàãðóæåíèå øòàìïà ñ äâóñâÿçíîé îáëàñòüþ êîí-òàêòàEccentric loading of the punch with doubly-connected contact domain 108

Ùåðáàêîâ Â.Â.Îá îäíîé âàðèàöèîííîé çàäà÷å ìåõàíèêè êîìïîçèöèîííûõ ìàòåðè-àëîâOn a variational problem arising in mechanics of composite materials 109

Ûñêàê Ò.Ê.Ýêñïîíåíöèàëüíàÿ óñòîé÷èâîñòü äèôôåðåíöèàëüíûõ óðàâíåíèé ñçàïàçäûâàþùèì àðãóìåíòîìExponential stability of delay dierential equations . . . . . . . . . . . . . . . . . . 110

Abiev N.A.On long time behavior of solutions of the normalized Ricci ow onspecial generalized Wallach spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

Antontsev S.N.A class of electromagnetic p(x, t)-curl systems: existence and unique-ness, blow-up and nite time extinction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

Apanasov B.N.Topological barriers for locally homeomorphic quasiregular mappingsin 3-space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

Assanova A.T.On the unique solvability of periodic problem for the Sobolev-typepartial dierential equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

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Chirkunov Yu.A.Conformal invariance of the elastostatics equations . . . . . . . . . . . . . . . . . . 116

Chirkunov Yu.A., Pikmullina E.O.Invariant modeling of the model of thermal motion of gas in a rareedspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

Ferone A., Korobkov M.V., Roviello A.On Luzin N-property for Sobolev mappings . . . . . . . . . . . . . . . . . . . . . . . . . 119

Golubyatnikov V.P., Kirillova N. E.On models of circular gene networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

Gutman A.E., Kononenko L. I.Binary correspondences and problems of chemical kinetics with many-sheeted slow surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

Isangulova D.V.Criterion for the potentiality of a vector eld in the sub-Riemanniangeometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

Jain P.Hardy type inequalities for monotone and quasi-monotone functions . 124

Kanjilal S.Linear canonical transform connected with various convolutions . . . . . 125

Kumar R.Zayed type fractional convolutions and distributions . . . . . . . . . . . . . . . . . 126

Kuznetsov I.V., Sazhenkov S.A.Entropy solutions of genuinely nonlinear forward-backward ultra-parabolic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

Singh M.Duality of some generalized grand Lebesgue spaces . . . . . . . . . . . . . . . . . . 128

Temirbekova L.N.Discretization of two-dimensional GelfandLevitan integral equation . 129

Vasilyeva O. I.LotkaVolterra competition models in advective environments . . . . . . . 131

Vodopyanov S.K.Sobolev spaces and quasi-conformal mappings in (sub)-Riemanniangeometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

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ÖÈÊËÛ ËÅÊÖÈÉ

Òåçèñû

LECTURE COURSES

Abstracts

Ìåæäóíàðîäíàÿ øêîëà-êîíôåðåíöèÿ Ñîáîëåâñêèå ÷òåíèÿ

Î ÒÎÏÎËÎÃÈ×ÅÑÊÎÉ ÊËÀÑÑÈÔÈÊÀÖÈÈÄÈÍÀÌÈ×ÅÑÊÈÕ ÑÈÑÒÅÌ ÍÀ

ÌÍÎÃÎÎÁÐÀÇÈßÕ ÐÀÇÌÅÐÍÎÑÒÈ 2 È 3

Ãðèíåñ Â. Ç.

Íàöèîíàëüíûé èññëåäîâàòåëüñêèé óíèâåðñèòåò

Âûñøàÿ øêîëà ýêîíîìèêè, Íèæíèé Íîâãîðîä, Ðîññèÿ;

[email protected]

Ìèíè-êóðñ ïîñâÿùåí èçëîæåíèþ ðåçóëüòàòîâ ïî òîïîëîãè÷åñêîéêëàññèôèêàöèè ñòðóêòóðíî óñòîé÷èâûõ äèñêðåòíûõ äèíàìè÷åñêèõ ñèñ-òåì (êàñêàäîâ), çàäàííûõ íà çàìêíóòûõ ãëàäêèõ ìíîãîîáðàçèÿõ ðàçìåð-íîñòè ìåíüøåé èëè ðàâíîé òðåì. Äëÿ çíàêîìñòâà ñ òåìàòèêîé, ññûëêàìèíà èñòî÷íèêè è îñíîâíûìè ðåçóëüòàòàìè ïîëåçíî îáðàòèòüñÿ ê ñòàòüÿì,îáçîðàì è ìîíîãðàôèÿì [15].

Ïåðâûé êëàññèôèêàöèîííûé ðåçóëüòàò áûë ïîëó÷åí À. Ã. Ìàéåðîì â1939 ã. äëÿ ñòðóêòóðíî óñòîé÷èâûõ (ãðóáûõ) äèôôåîìîðôèçìîâ îêðóæ-íîñòè. Îí ïîêàçàë, ÷òî íåáëóæäàþùåå ìíîæåñòâî òàêîãî äèôôåîìîð-ôèçìà ñîñòîèò èç êîíå÷íîãî ÷èñëà ãèïåðáîëè÷åñêèõ ïåðèîäè÷åñêèõ òî-÷åê è äàë ïîëíóþ òîïîëîãè÷åñêóþ êëàññèôèêàöèþ ãðóáûõ äèôôåîìîð-ôèçìîâ îêðóæíîñòè.

Áëàãîäàðÿ Ñ. Ñìåéëó è Ä.Â. Àíîñîâó â 60-õ ãîäàõ ïðîøëîãî âå-êà ñòàëî ÿñíî, ÷òî íåáëóæäàþùåå ìíîæåñòâî ñòðóêòóðíî óñòîé÷èâîãîäèôôåîìîðôèçìà ïîâåðõíîñòè ìîæåò èìåòü ñ÷åòíîå ïîäìíîæåñòâî ïå-ðèîäè÷åñêèõ òî÷åê. Ïîýòîìó ñòðóêòóðíî óñòîé÷èâûå êàñêàäû, íåáëóæ-äàþùèå ìíîæåñòâà êîòîðûõ êîíå÷íû, áûëè îïðåäåëåíû ñïåöèàëüíûìîáðàçîì è ïîëó÷èëè íàçâàíèå êàñêàäîâ Ìîðñà Ñìåéëà.

Ïîëíûé òîïîëîãè÷åñêèé èíâàðèàíò äëÿ äèôôåîìîðôèçìîâ Ìîðñà Ñìåéëà íàìíîãî ñëîæíåå, ÷åì â îäíîìåðíîé ñèòóàöèè. Îí áûë ïîëó÷åíâ ñåðèè ðàáîò Õ. Áîíàòòè, À.Í. Áåçäåíåæíûõ, Â. Ç. Ãðèíåñà, Ð. Ëàíæå-âåíà è äð. â 80-õ 90-õ ãîäàõ ïðîøëîãî ñòîëåòèÿ. Ýòîò èíâàðèàíò ïðåä-ñòàâëÿåò èç ñåáÿ íåêîòîðûé ãðàô, îñíàùåííûé àâòîìîðôèçìîì è äîïîë-íèòåëüíîé èíôîðìàöèåé, îïèñûâàþùåé ñòðóêòóðó ìíîæåñòâà ïåðåñå÷å-íèÿ óñòîé÷èâûõ è íåóñòîé÷èâûõ ìíîãîîáðàçèé ñåäëîâûõ ïåðèîäè÷åñêèõòî÷åê. Â íåäàâíåå âðåìÿ àâòîðîì äîêëàäà ñîâìåñòíî ñ Î.Â. Ïî÷èíêîé èÑ. Âàí Ñòðèíîì ïðåäëîæåí íîâûé ãåîìåòðè÷åñêèé ïîäõîä, êîòîðûé ïîç-âîëÿåò ðåøèòü ïðîáëåìó ðåàëèçàöèè, ò. å. ïîñòðîåíèÿ â êàæäîì êëàññåòîïîëîãè÷åñêîé ñîïðÿæåííîñòè ñòàíäàðòíîãî ïðåäñòàâèòåëÿ.

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Ñîãëàñíî ñïåêòðàëüíîé òåîðåìå Ñ. Ñìåéëà íåáëóæäàþùåå ìíîæå-ñòâî ñòðóêòóðíî óñòîé÷èâîãî êàñêàäà f , ñîäåðæàùåãî áåñêîíå÷íîå ìíî-æåñòâî ïåðèîäè÷åñêèõ òî÷åê, ñîñòîèò èç êîíå÷íîãî ÷èñëà çàìêíóòûõèíâàðèàíòíûõ ìíîæåñòâ, êàæäîå èç êîòîðûõ ñîäåðæèò òðàíçèòèâíóþîðáèòó. Ýòè ìíîæåñòâà íàçûâàþòñÿ áàçèñíûìè. Ñâîéñòâà äèôôåîìîð-ôèçìà f âî ìíîãîì îïðåäåëÿþòñÿ äèíàìèêîé îãðàíè÷åíèÿ äèôôåîìîð-ôèçìà íà áàçèñíûå ìíîæåñòâà.• Åñëè òîïîëîãè÷åñêàÿ ðàçìåðíîñòü áàçèñíîãî ìíîæåñòâà Ω ñòðóê-

òóðíî óñòîé÷èâîãî äèôôåîìîðôèçìà f : M2 → M2 ïîâåðõíîñòè M2

ðàâíà 2, òî íåñóùàÿ ïîâåðõíîñòü M2 ÿâëÿåòñÿ äâóìåðíûì òîðîì, äèô-ôåîìîðôèçì f ÿâëÿåòñÿ äèôôåîìîðôèçìîì Àíîñîâà è òîïîëîãè÷åñêèñîïðÿæåí ñ àâòîìîðôèçìîì òîðà, èíäóöèðîâàííûì ãèïåðáîëè÷åñêîéóíèìîäóëÿðíîé ìàòðèöåé (ß. Ñèíàé, 1968).• Åñëè ðàçìåðíîñòü áàçèñíîãî ìíîæåñòâà Ω ñòðóêòóðíî óñòîé÷èâîãî

äèôôåîìîðôèçìà f : M2 →M2 ðàâíà åäèíèöå, òî îíî ÿâëÿåòñÿ àòòðàê-òîðîì èëè ðåïåëëåðîì è ëîêàëüíî ãîìåîìîðôíî ïðÿìîìó ïðîèçâåäåíèþèíòåðâàëà è êàíòîðîâñêîãî ìíîæåñòâà. Äèíàìèêà îãðàíè÷åíèÿ äèôôåî-ìîðôèçìîâ f íà îäíîìåðíîå áàçèñíîå ìíîæåñòâî ïîëíîñòüþ îïðåäåëÿ-åòñÿ èíäóöèðîâàííûì àâòîìîðôèçìîì ôóíäàìåíòàëüíîé ãðóïïû íîñè-òåëÿ áàçèñíîãî ìíîæåñòâà. Áîëåå òîãî, îãðàíè÷åíèå äèôôåîìîðôèçìàíà íîñèòåëü òîïîëîãè÷åñêè ñîïðÿæåíî ñ äèíàìèêîé äèôôåîìîðôèçìà,ïîëó÷åííîãî èç àíîñîâñêîãî èëè ïñåâäîàíîñîâñêîãî äèôôåîìîðôèçìàïîñðåäñòâîì õèðóðãè÷åñêîé îïåðàöèè, ïðåäëîæåííîé Ñ. Ñìåéëîì. Îïè-ñàííûå ðåçóëüòàòû áûëè ïîëó÷åíû Â. Ç. Ãðèíåñîì, Ð.Â. Ïëûêèíûì,À.Þ. Æèðîâûì, Õ.Õ. Êàëàåì è äð. â 70-õ 90-õ ãîäàõ ïðîøëîãî âåêà.• Åñëè ðàçìåðíîñòü íåòðèâèàëüíîãî (îòëè÷íîãî îò ïåðèîäè÷åñêîé

îðáèòû) áàçèñíîãî ìíîæåñòâà ðàâíà íóëþ, òî îãðàíè÷åíèå äèôôåîìîð-ôèçìà f íà áàçèñíîå ìíîæåñòâî òîïîëîãè÷åñêè ñîïðÿæåíî ñî ñäâèãîìíåêîòîðîé òîïîëîãè÷åñêîé öåïè Ìàðêîâà, è âîïðîñ îá îïèñàíèè îãðà-íè÷åíèÿ äèôôåîìîðôèçìà íà îêðåñòíîñòü íóëüìåðíîãî áàçèñíîãî ìíî-æåñòâà òðåáóåò äîïîëíèòåëüíîé èíôîðìàöèè. Ýòà ïðîáëåìà ðåøàëàñüïðè ðàçëè÷íûõ ïðåäïîëîæåíèÿõ â ðàáîòàõ Õ. Áîíàòòè, Â. Ç. Ãðèíåñà,Õ.Õ. Êàëàÿ, Ð. Ëàíæåâåíà â 80-õ 90-õ ãîäàõ ïðîøëîãî âåêà.

Èíôîðìàöèÿ î äèíàìèêå îãðàíè÷åíèÿ äèôôåîìîðôèçìà íà îêðåñò-íîñòè íåòðèâèàëüíûõ áàçèñíûõ ìíîæåñòâ è ðåçóëüòàòû ïî êëàññèôèêà-öèè äèôôåîìîðôèçìîâ Ìîðñà Ñìåéëà áûëè ïðèìåíåíû â 70-õ 90-õãîäàõ ïðîøëîãî âåêà äëÿ ïîëó÷åíèÿ ïîëíîé êëàññèôèêàöèè ñòðóêòóðíîóñòîé÷èâûõ êàñêàäîâ íà ïîâåðõíîñòÿõ ñ îäíîìåðíûìè áàçèñíûìè ìíî-

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æåñòâàìè è íóëüìåðíûìè áàçèñíûìè ìíîæåñòâàìè áåç ñîïðÿæåííûõòî÷åê (Â. Ç. Ãðèíåñ, À.Þ. Æèðîâ, Õ.X. Êàëàé, Ð.Â. Ïëûêèí), à òàê-æå äëÿ ïðîèçâîëüíûõ ñòðóêòóðíî óñòîé÷èâûõ êàñêàäîâ (Õ. Áîíàòòè,Ð. Ëàíæåâåí).

Â 2000 ãîäó ñòàëî ÿñíî, ÷òî äëÿ òðåõìåðíûõ êàñêàäîâ Ìîðñà Ñìåé-ëà íå äîñòàòî÷íî êîìáèíàòîðíûõ èíâàðèàíòîâ. Õ. Áîíàòòè, Â. Ç. Ãðè-íåñîì, Ä. Ïèêñòîíîì áûëî îáíàðóæåíî, ÷òî ñóùåñòâóåò ñ÷åòíîå ìíîæå-ñòâî íå ñîïðÿæåííûõ äèôôåîìîðôèçìîâ Ìîðñà Ñìåéëà ñ íåáëóæäà-þùèì ìíîæåñòâîì, ñîñòîÿùèì ðîâíî èç ÷åòûðåõ íåïîäâèæíûõ òî÷åê(ñåäëîâîé òî÷êè, äâóõ ñòîêîâ è îäíîãî èñòî÷íèêà). Ýòî ÿâëåíèå îáúÿñ-íÿåòñÿ âîçìîæíîñòüþ äèêîãî âëîæåíèÿ ñåïàðàòðèñ ñåäëîâîé íåïîäâèæ-íîé òî÷êè. Îáíàðóæåííûé ôåíîìåí ñòàë ñòèìóëîì ê ââåäåíèþ íîâûõòîïîëîãè÷åñêèõ èíâàðèàíòîâ, ñâÿçàííûõ ñ âëîæåíèåì óçëîâ, äâóìåðíûõòîðîâ è áóòûëîê Êëåéíà â ïðîñòðàíñòâî îðáèò îãðàíè÷åíèÿ äèôôåî-ìîðôèçìà Ìîðñà Ñìåéëà íà ñïåöèàëüíûì îáðàçîì âûáðàííîå ìíîæå-ñòâî áëóæäàþùèõ òî÷åê. Ýòè èíâàðèàíòû ïîçâîëèëè ïîëó÷èòü ê íàñòî-ÿùåìó âðåìåíè â ñåðèè ðàáîò Õ. Áîíàòòè, Â. Ç. Ãðèíåñà, Â.Ñ. Ìåäâåäå-âà, Ý. Ïåêó è Î.Â. Ïî÷èíêè ïîëíóþ òîïîëîãè÷åñêóþ êëàññèôèêàöèþêàñêàäîâ Ìîðñà Ñìåéëà íà 3-ìíîãîîáðàçèÿõ.

Ïàðàëëåëüíî, íà÷èíàÿ ñ 70-õ ãîäîâ ïðîøëîãî âåêà, áûëè íà÷àòû èñ-ñëåäîâàíèÿ ïî òîïîëîãè÷åñêîé êëàññèôèêàöèè ñòðóêòóðíî óñòîé÷èâûõêàñêàäîâ íà 3-ìíîãîîáðàçèÿõ, íåáëóæäàþùèå ìíîæåñòâà êîòîðûõ ñî-äåðæàò ñ÷åòíîå ìíîæåñòâî ïåðèîäè÷åñêèõ òî÷åê.

Â êà÷åñòâå èëëþñòðàöèè ñôîðìóëèðóåì íåñêîëüêî çàêîí÷åííûõ ðå-çóëüòàòîâ â ýòîì íàïðàâëåíèè, äàþùèõ ïðåäñòàâëåíèå î äîñòèãíóòîì êíàñòîÿùåìó âðåìåíè ïðîãðåññå.• Åñëè òîïîëîãè÷åñêàÿ ðàçìåðíîñòü áàçèñíîãî ìíîæåñòâà ñòðóêòóð-

íî óñòîé÷èâîãî äèôôåîìîðôèçìà f : M3 → M3 òðåõìåðíîãî ìíîãî-îáðàçèÿ M3 ðàâíà 3, òî M3 ÿâëÿåòñÿ òðåõìåðíûì òîðîì, è äèôôåî-ìîðôèçì f ÿâëÿåòñÿ äèôôåîìîðôèçìîì Àíîñîâà, òîïîëîãè÷åñêè ñî-ïðÿæåííûì ñ àòîìîðôèçìîì òîðà, èíäóöèðîâàííûì ãèïåðáîëè÷åñêîéóíèìîäóëÿðíîé ìàòðèöåé (Äæ. Ôðåíêñ, Ø. Íüþõàóñ, 1970).• Åñëè íåáëóæäàþùåå ìíîæåñòâî ñòðóêòóðíî óñòîé÷èâîãî äèôôåî-

ìîðôèçìà f : M3 →M3 ñîñòîèò èç áàçèñíûõ ìíîæåñòâ, òîïîëîãè÷åñêàÿðàçìåðíîñòü êîòîðûõ ðàâíà äâóì, òî:1) îãðàíè÷åíèå íåêîòîðîé ñòåïåíè äèôôåîìîðôèçìà f íà êàæäîå áàçèñ-íîå ìíîæåñòâî òîïîëîãè÷åñêè ñîïðÿæåíî àâòîìîðôèçìó Àíîñîâà äâó-ìåðíîãî òîðà;

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2) ìíîãîîáðàçèå M3 ãîìåîìîðôíî ôàêòîðïðîñòðàíñòâó, ïîëó÷åííîìóèç ïðÿìîãî ïðîèçâåäåíèÿ äâóìåðíîãî òîðà íà åäèíè÷íûé èíòåðâàë ïðèîòîæäåñòâëåíèè ãðàíèö ïðîèçâåäåíèÿ ïîñðåäñòâîì íåêîòîðîãî ãîìåî-ìîðôèçìà;3) íàéäåí ïîëíûé òîïîëîãè÷åñêèé èíâàðèàíò êàñêàäîâ îïèñàííîãî êëàñ-ñà, è ðåøåíà çàäà÷à ðåàëèçàöèè, ò. å. â êàæäîì êëàññå òîïîëîãè÷åñêîéñîïðÿæåííîñòè ïîñòðîåí ñòàíäàðòíûé ïðåäñòàâèòåëü(Â. Ç. Ãðèíåñ, Â.Ñ. Ìåäâåäåâ, ß.À. Ëåâ÷åíêî, Î.Â. Ïî÷èíêà, 20052015).• Ïóñòü f : M3 → M3 ñòðóêòóðíî óñòîé÷èâûé äèôôåîìîðôèçì,

íåáëóæäàþùåå ìíîæåñòâî êîòîðîãî ñîäåðæèò äâóìåðíûé ðàñòÿãèâàþ-ùèéñÿ àòòðàêòîð. Òîãäà ìíîãîîáðàçèå M3 ãîìåîìîðôíî òðåõìåðíîìóòîðó, à f òîïîëîãè÷åñêè ñîïðÿæåí ñ äèôôåîìîðôèçìîì, ïîëó÷åííûìèç äèôôåîìîðôèçìà Àíîñîâà, ïîñðåäñòâîì îáîáùåííîé õèðóðãè÷åñêîéîïåðàöèè Ñ. Ñìåéëà (Â. Ç. Ãðèíåñ, Å.Â. Æóæîìà, 20022005).

Ëåêöèè ïîäãîòîâëåíû ïðè ïîääåðæêå ÐÔÔÈ (ïðîåêòû 15-01-03687-a,

16-51-10005-Ko_a) è â ðàìêàõ ïðîãðàììû ôóíäàìåíòàëüíûõ èññëåäîâàíèé

ÍÈÓ ÂØÝ â 2017 ãîäó (ïðîåêò 90).

ËÈÒÅÐÀÒÓÐÀ

1. Àðàíñîí Ñ.Õ., Ãðèíåñ Â. Ç. Òîïîëîãè÷åñêàÿ êëàññèôèêàöèÿ êàñêàäîâ íàçàìêíóòûõ äâóìåðíûõ ìíîãîîáðàçèÿõ // Óñïåõè ìàò. íàóê. 1990. Ò. 45,âûï. 1. Ñ. 332. (Àíãë. ïåðåâîä: Aranson S.Kh., Grines V. Z. The topologicalclassication of cascades on closed two-dimensional manifolds // Russ. Math.Surv. 1990. V. 45, No 1. P. 135.)

2. Ãðèíåñ Â. Ç., Ïî÷èíêà Î.Â. Ââåäåíèå â òîïîëîãè÷åñêóþ êëàññèôèêàöèþêàñêàäîâ íà ìíîãîîáðàçèÿõ ðàçìåðíîñòè äâà è òðè. Ìîñêâà, Èæåâñê: Ðå-ãóëÿðíàÿ è õàîòè÷åñêàÿ äèíàìèêà, 2011.

3. Ãðèíåñ Â. Ç., Ïî÷èíêà Î.Â. Êàñêàäû Ìîðñà Ñìåéëà íà 3-ìíîãîîáðàçè-ÿõ // Óñïåõè ìàò. íàóê. 2013. Ò. 68, âûï. 1. Ñ. 129188. (Àíãë. ïåðåâîä:Grines V. Z., Pochinka O.V. MorseSmale cascades on 3-manifolds // Russ.Math. Surv. 2013. V. 68, No 1. P. 117173.)

4. Grines V., Levchenko Yu., Medvedev V., Pochinka O. The topologicalclassication of structurally stable 3-dieomorphisms with two-dimensionalbasic sets // Nonlinearity. 2015. V. 28, No 11. P. 40814102.

5. Grines V., Medvedev T., Pochinka O. Dynamical systems on 2- and 3-mani-folds (Developments in Mathematics, V. 46). Cham (Switzerland): Springer,2016.

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ÎÁ ÎÄÍÎÌ ÍÎÂÎÌ ÌÅÒÎÄÅ ÈÑÑËÅÄÎÂÀÍÈßÑËÀÁÎÉ ÐÀÇÐÅØÈÌÎÑÒÈ ÇÀÄÀ×

ÃÈÄÐÎÄÈÍÀÌÈÊÈ

Çâÿãèí Â. Ã.

Âîðîíåæñêèé ãîñóäàðñòâåííûé óíèâåðñèòåò, Âîðîíåæ, Ðîññèÿ;

[email protected]

Â ëåêöèÿõ ïðåäïîëàãàåòñÿ:1. Èçëîæèòü ðàçíûå âàðèàíòû àïïðîêñèìàöèîííî-òîïîëîãè÷åñêîãî

ìåòîäà èññëåäîâàíèÿ ñëàáîé ðàçðåøèìîñòè íà÷àëüíî-êðàåâûõ çàäà÷ãèäðîäèíàìèêè íà ïðèìåðå ñèñòåìû Íàâüå Ñòîêñà.

2. Èçëîæèòü òåîðèþ àòòðàêòîðîâ àâòîíîìíûõ è íåàâòîíîìíûõ óðàâ-íåíèé ãèäðîäèíàìèêè, îñíîâàííóþ íà ïîíÿòèè íåèíâàðèàíòíîãî òðàåê-òîðíîãî ïðîñòðàíñòâà.

3. Èçëîæèòü íà ïðèìåðàõ ìåòîä èññëåäîâàíèÿ îïòèìàëüíûõ çàäà÷ ñîáðàòíîé ñâÿçüþ äëÿ óðàâíåíèé ãèäðîäèíàìèêè, îñíîâàííûé íà òîïî-ëîãè÷åñêîé òåîðèè ñòåïåíè ìíîãîçíà÷íûõ îòîáðàæåíèé áàíàõîâûõ ïðî-ñòðàíñòâ.


1. Çâÿãèí Â. Ã., Äìèòðèåíêî Â.Ò. Àïïðîêñèìàöèîííî-òîïîëîãè÷åñêèé ïîä-õîä ê èññëåäîâàíèþ çàäà÷ ãèäðîäèíàìèêè. Ñèñòåìà Íàâüå Ñòîêñà. Ì.:Åäèòîðèàë ÓÐÑÑ, 2004.

2. Zvyagin V.G. On solvability of some initial-boundary problems for mathema-tical models of the motion of nonlinearly viscous and viscoelastic uids //J. Math. Sci., New York. 2004. V. 124, No 5. P. 53215334.

3. Zvyagin V.G., Vorotnikov D.A. Topological approximation methods for evolu-tionary problems of nonlinear hydrodynamics (de Gruyter Series in NonlinearAnalysis and Applications, V. 12). Berlin: de Gruyter, 2008.

4. Çâÿãèí Â. Ã., Òóðáèí Ì.Â. Ìàòåìàòè÷åñêèå âîïðîñû ãèäðîäèíàìèêè âÿç-êîóïðóãèõ ñðåä. Ì.: ÊÐÀÑÀÍÄ, 2012.

5. Zvyagin V.G. Topological approximation approach to study of mathematicalproblems of hydrodynamics // J. Math. Sci., New York. 2014. V. 201, No 6.P. 830858.

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MATHEMATICS OF FLUIDS IN MOTION

Feireisl E.

Institute of Mathematics of the Academy of Sciences

of the Czech Republic, Prague, Czech Republic;

[email protected]

This is a series of lectures on recent development of the mathematicaltheory of compressible uids, both viscous and inviscid.

1. We start by reviewing some results on existence of solutions of theinitial/boundary value problem for the compressible NavierStokes andEuler system:

• Existence theory in the framework of regular (smooth) solutions.Local in time existence for arbitrary data and global existence forsmall data in the viscous case.

• Existence of weak (distributional) solutions for the NavierStokessystem.

• Existence of weak (distributional) solutions for the (barotropic) Eulersystem.

Recommended preliminary reading: [1, 2].

2. We discuss some recent results on ill/posedness of the Euler systemin the class of weak solutions:

• Method of convex integration and its adaptation to problems in uidmechanics.

• Existence of innitely many global-in-time weak solutions to thecompressible Euler system.

• Innitely many weak solutions that comply with entropy criteria.Solutions emanating from regular initial data.


3. Finally, we introduce the concept of oscillatory (dissipative) measure-valued solution and discuss some applications:

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• Dissipative measure-valued solutions.

• Weak-strong uniqueness.

• Applications to singular limits and numerical analysis.


The research of E.F. leading to these results has received funding from the

European Research Council under the European Union's Seventh Framework

Programme (FP7/20072013)/ ERC Grant Agreement 320078. The Institute of

Mathematics of the Academy of Sciences of the Czech Republic is supported by

RVO:67985840.

REFERENCES

1. Lions P.-L., Mathematical Topics in Fluid Mechanics, Vol. 2: CompressibleModels (Oxford Lecture Series in Mathematics and its Applications, Vol. 10),Clarendon Press, Oxford (1998).

2. Feireisl E., Dynamics of Viscous Compressible Fluids (Oxford Lecture Series inMathematics and its Applications, Vol. 26), Oxford University Press, Oxford(2004).

3. De Lellis C., Szekelyhidi L. (jr.), On admissibility criteria for weak solutions ofthe Euler equations, Arch. Ration. Mech. Anal., 195, No. 1, 225260 (2010).

4. Chiodaroli E., De Lellis C., Kreml O., Global ill-posedness of the isentropicsystem of gas dynamics, Commun. Pure Appl. Math., 68, No. 7, 11571190(2015).

5. Gwiazda P., Swierczewska-Gwiazda A., Wiedemann E., Weak-strong unique-ness for measure-valued solutions of some compressible uid models, Non-linearity, 28, No. 11, 38733890 (2015).

6. Feireisl E., Gwiazda P., Swierczewska-Gwiazda A., Wiedemann E., Dissipativemeasure-valued solutions to the compressible NavierStokes system, Calc.Var. Partial Dier. Equ., 55, No. 6, Article ID 141 (2016).

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Î ÏÅÐÂÎÉ ÍÀÓ×ÍÎÉ ÐÀÁÎÒÅ Ñ.Ë. ÑÎÁÎËÅÂÀ

×å÷êèíà À. Ã.

Ìîñêîâñêèé ãîñóäàðñòâåííûé óíèâåðñèòåò èì. Ì.Â. Ëîìîíîñîâà,

Ìîñêâà, Ðîññèÿ; [email protected]

Â ëåêöèÿõ áóäåò ðàññêàçàíî î ïåðâîé íàó÷íîé ðàáîòå Ñåðãåÿ Ëüâî-âè÷à Ñîáîëåâà [1]. Ýòà ðàáîòà áûëà íàïèñàíà â ñòóäåí÷åñêèå ãîäû. Ïî-âîäîì ïîñëóæèëè ðåçóëüòàòû Í.Í. Ñàëòûêîâà, î êîòîðûõ ðàññêàçûâàëïðîôåññîð Í.Ì. Ãþíòåð íà ëåêöèÿõ ïî óðàâíåíèÿì ñ ÷àñòíûìè ïðîèç-âîäíûìè. Íåêîòîðûå èõ íèõ âûçâàëè ñîìíåíèÿ ó ñòóäåíòà Ñ. Ñîáîëå-âà. Ïûòàÿñü ðàçîáðàòüñÿ â ðàáîòàõ Í.Í. Ñàëòûêîâà, Ñ. Ñîáîëåâ íàøåëíåòî÷íîñòè è ïîñòðîèë êîíòðïðèìåðû.


1. Ñîáîëåâ Ñ.Ë. Çàìå÷àíèå ïî ïîâîäó ðàáîò ïðîô. Ñàëòûêîâà Èññëåäîâà-íèÿ ïî òåîðèè óðàâíåíèé ñ ÷àñòíûìè ïðîèçâîäíûìè ïåðâîãî ïîðÿäêà îä-íîé íåèçâåñòíîé ôóíêöèè è Î ðàçâèòèè òåîðèè óðàâíåíèé ñ ÷àñòíûìèïðîèçâîäíûìè ïåðâîãî ïîðÿäêà îäíîé íåèçâåñòíîé ôóíêöèè. Ðóêîïèñü.1927. Ñåìåéíûé àðõèâ.

26

ÑÅÊÖÈÎÍÍÛÅÄÎÊËÀÄÛ

Òåçèñû

SHORTCOMMUNICATIONS

Abstracts


ËÈÍÅÉÍÀß ÎÁÐÀÒÍÀß ÇÀÄÀ×À ÄËßÎÏÅÐÀÒÎÐÍÎ-ÄÈÔÔÅÐÅÍÖÈÀËÜÍÎÃÎÓÐÀÂÍÅÍÈß ÑÌÅØÀÍÍÎÃÎ ÒÈÏÀ

Ñ ÏÀÐÀÌÅÒÐÎÌ

Àáàøååâà Í.Ë.

Èíñòèòóò ìàòåìàòèêè èì. Ñ.Ë. Ñîáîëåâà ÑÎ ÐÀÍ,Íîâîñèáèðñêèé ãîñóäàðñòâåííûé óíèâåðñèòåò, Íîâîñèáèðñê, Ðîññèÿ;

[email protected]

Ðàññìàòðèâàåòñÿ ëèíåéíàÿ îáðàòíàÿ çàäà÷à: íàéòè ôóíêöèè u(t, p)è ϕ(t), óäîâëåòâîðÿþùèå óðàâíåíèþ

But + pLu = ϕ(t) + f(t, p), p ∈ D, t ∈ (0, T ), (1)

è êðàåâûì óñëîâèÿìu(0, p) = u(T, p) = 0.

Çäåñü T < ∞, D ⊂ C èçìåðèìîå ìíîæåñòâî, èìåþùåå ïðåäåëüíóþòî÷êó p0 ∈ C, L, B ñàìîñîïðÿæåííûå îïåðàòîðû â äàííîì êîìïëåêñ-íîì ãèëüáåðòîâîì ïðîñòðàíñòâå E ñî ñêàëÿðíûì ïðîèçâåäåíèåì (·, ·) èíîðìîé ‖ · ‖. Îïåðàòîð B èìååò ïðîèçâîëüíîå ðàñïîëîæåíèå ñïåêòðà, àñïåêòð îïåðàòîðà L ïîëóîãðàíè÷åí.

Â ðàáîòå íà îñíîâå èçâåñòíûõ ðåçóëüòàòîâ äëÿ ïðÿìûõ çàäà÷ èññëå-äîâàíà îáðàòíàÿ çàäà÷à íàõîæäåíèÿ ñâîáîäíîãî ÷ëåíà äëÿ óðàâíåíèÿ ñïàðàìåòðîì (1). Çäåñü äîêàçàòåëüñòâî ïðîâîäèòñÿ ìåòîäîì ðàçëîæåíèÿâ ðÿä ïî ñîáñòâåííûì è ïðèñîåäèíåííûì ýëåìåíòàì ñîîòâåòñòâóþùåéñïåêòðàëüíîé çàäà÷è

Lu = λBu,

è äàíî ÿâíîå ïðåäñòàâëåíèå ðåøåíèÿ.Ðàáîòà âûïîëíåíà ïðè ïîääåðæêå Ðîññèéñêîãî ôîíäà ôóíäàìåíòàëüíûõ

èññëåäîâàíèé (ïðîåêò 15-01-06582).

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ÎÁÙÅÅ ÐÅØÅÍÈÅ ÇÀÄÀ×ÈÌÈÍÈÌÈÇÀÖÈÈ ÐÀÑÕÎÄÀ ÐÅÑÓÐÑÀ

Àëåêñàíäðîâ Â.Ì.

Èíñòèòóò ìàòåìàòèêè èì. Ñ.Ë. Ñîáîëåâà ÑÎ ÐÀÍ,Íîâîñèáèðñê, Ðîññèÿ; [email protected]

Ïóñòü óïðàâëÿåìûé îáúåêò îïèñûâàåòñÿ ëèíåéíûì äèôôåðåíöèàëü-íûì óðàâíåíèåì x = A(t)x + B(t)u, x(t0) = x0, x0 ∈ D, ãäå u m-ìåðíûé âåêòîð óïðàâëåíèÿ, êîìïîíåíòû êîòîðîãî ïîä÷èíåíû îãðàíè÷å-íèÿì |uj | 6 Mj , j = 1,m. Ïðåäïîëàãàåòñÿ, ÷òî ñèñòåìà ïîêîìïîíåíòíîïîëíîñòüþ óïðàâëÿåìà è ïåðåâîäèìà â çàäàííîå êîíå÷íîå ñîñòîÿíèå èçîãðàíè÷åííîé îáëàñòè íà÷àëüíûõ óñëîâèé D.

Çàäà÷à. Íàéòè äîïóñòèìîå óïðàâëåíèå u0(t), ïåðåâîäÿùåå çà ôèê-ñèðîâàííîå âðåìÿ T = tk − t0 (ãäå T > T0) ñèñòåìó èç íà÷àëüíîãî ñî-ñòîÿíèÿ x(t0) = x0 â êîíå÷íîå ñîñòîÿíèå x(tk) = xk è ìèíèìèçèðóþùåå

ôóíêöèîíàë J(u) =tk∫t0

∑mj=1 |uj(τ)|dτ . Çäåñü T0 âðåìÿ îïòèìàëüíîãî

ïî áûñòðîäåéñòâèþ ïåðåâîäà ñèñòåìû.Ðàçðàáîòàí îáùèé ìåòîä âû÷èñëåíèÿ îïòèìàëüíîãî ïî ðàñõîäó ðå-

ñóðñà óïðàâëåíèÿ, îñíîâàííûé íà ðàçäåëåíèè çàäà÷è íà äâå íåçàâèñè-ìûå ïîäçàäà÷è: 1) âû÷èñëåíèå ñòðóêòóðû îïòèìàëüíîãî óïðàâëåíèÿ;2) âû÷èñëåíèå ìîìåíòîâ ïåðåêëþ÷åíèé îïòèìàëüíîãî óïðàâëåíèÿ. Âû-÷èñëåíèå ñòðóêòóðû îñíîâàíî íà îðèãèíàëüíîì ìåòîäå ôîðìèðîâàíèÿêâàçèîïòèìàëüíîãî óïðàâëåíèÿ. Âû÷èñëåíèå ìîìåíòîâ ïåðåêëþ÷åíèéóïðàâëåíèÿ îñíîâàíî íà íàéäåííîé ñâÿçè ìåæäó îòêëîíåíèÿìè íà÷àëü-íûõ óñëîâèé ñîïðÿæåííîé ñèñòåìû ñ îòêëîíåíèÿìè ôàçîâîé òðàåêòîðèèâ êîíå÷íûé ìîìåíò. Äàí ìåòîä çàäàíèÿ íà÷àëüíîãî ïðèáëèæåíèÿ. Ðàç-ðàáîòàí èòåðàöèîííûé àëãîðèòì è ðàññìîòðåíû åãî îñîáåííîñòè. Ïðè-âåäåíû ðåçóëüòàòû ìîäåëèðîâàíèÿ è ÷èñëåííûõ ðàñ÷åòîâ.

Ðàáîòà âûïîëíåíà ïðè ôèíàíñîâîé ïîääåðæêå Ðîññèéñêîãî ôîíäà ôóíäà-

ìåíòàëüíûõ èññëåäîâàíèé (ïðîåêò 16-01-00592à).


1. Àëåêñàíäðîâ Â.Ì. Âûðîæäåííîå ðåøåíèå çàäà÷è ìèíèìèçàöèè ðàñõîäàðåñóðñà // Ñèá. æóðí. âû÷èñë. ìàòåìàòèêè. 2016. Ò. 19, 1. Ñ. 518.

2. Àëåêñàíäðîâ Â.Ì. Êâàçèîïòèìàëüíîå óïðàâëåíèå äèíàìè÷åñêèìè ñèñòå-ìàìè // Àâòîìàòèêà è òåëåìåõàíèêà. 2016. Âûï. 7. Ñ. 4767.

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ÌÅÒÎÄ ÏËÎÑÊÈÕ ÑÐÅÄÍÈÕ ÄËßÃÈÏÅÐÁÎËÈ×ÅÑÊÈÕ ÑÈÑÒÅÌ

Àíèêîíîâ Ä.Ñ.


Ðàññìîòðèì çàäà÷ó î êîëåáàíèÿõ îäíîðîäíîé íåîãðàíè÷åííîé ìåì-áðàíû ïðè íàëè÷èè âíåøíåé ñèëû. Ñîîòâåòñòâóþùåå óðàâíåíèå êîëå-áàíèé èìååò âèä

utt − a2(ux1x1+ ux2x2

) = f(t, x1, x2), (x1, x2) ∈ R2, t > 0, (1)

ãäå f ∈ C1((0,∞)×R2), a ∈ R, a > 0. Ïóñòü çàäàíû íà÷àëüíûå äàííûå:

u(0, x1, x2) = ϕ(x1, x2), ut(0, x1, x2) = ψ(x1, x2),

ϕ(x1, x2) ∈ C2(R2), ψ(x1, x2) ∈ C1(R2).

Ðàññìîòðèì ñëåäóþùèå îáðàòíûå çàäà÷è. Äëÿ ýòîãî ïðåäïîëîæèì,÷òî ôóíêöèÿ f íå çàâèñèò îò t, ò. å. f(t, x1, x2) = f1(x1, x2), f1 ∈ C1(R2),ïðè÷åì f1 ìîæåò ïðèíèìàòü ëèøü íåîòðèöàòåëüíûå çíà÷åíèÿ, à ìíîæå-ñòâî òî÷åê, â êîòîðûõ ôóíêöèÿ f1 ïîëîæèòåëüíà, ÿâëÿåòñÿ íåêîòîðîéîãðàíè÷åííîé îáëàñòüþ G. Ôóíêöèè ϕ, ψ ïðåäïîëàãàþòñÿ ôèíèòíûìè.

Çàäà÷à 1. Èìåÿ â êà÷åñòâå èñõîäíûõ äàííûõ êîýôôèöèåíò a, ôóíê-öèè ϕ, ψ, u(t, x1, 0), ux2

(t, x1, 0), u(t, 0, x2), ux1(t, 0, x2), íàéòè òàêîé ïðÿ-

ìîóãîëüíèê E, ñîäåðæàùèé â ñåáå îáëàñòü G, ÷òîáû êàæäàÿ åãî ñòîðîíàêàñàëàñü ãðàíèöû îáëàñòè G.

Çàäà÷à 2. Èìåÿ â êà÷åñòâå èñõîäíûõ äàííûõ èíòåðâàë [a1, a2] äëÿêîýôôèöèåíòà a, ôóíêöèè u(t, x1, 0), ux2

(t, x1, 0), u(t, 0, x2), ux1(t, 0, x2)

è ïîëàãàÿ ϕ = 0, ψ = 0, íàéòè ïðÿìîóãîëüíèê E, ñîäåðæàùèé â ñåáåîáëàñòü G.

Ïîäðàçóìåâàåòñÿ, ÷òî ðàçìåðû èñêîìîãî ïðÿìîóãîëüíèêà E, ïî âîç-ìîæíîñòè, äîëæíû áûòü ìèíèìàëüíû.

Ñìûñë ýòèõ çàäà÷ ñîñòîèò â ëîêàëèçàöèè íîñèòåëÿ ïðàâîé ÷àñòèóðàâíåíèÿ (1), ÷òî ñ ôèçè÷åñêîé òî÷êè çðåíèÿ îçíà÷àåò ïðèáëèæåííîåîïðåäåëåíèå ìåñòà âîçäåéñòâèÿ âíåøíåé ñèëû íà ìåìáðàíó.

Äëÿ îáåèõ çàäà÷ ïîñòðîåíû è òåñòèðîâàíû àëãîðèòìû.

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ËÓ×ÅÂÎÉ ÌÅÒÎÄ È ÂÎÏÐÎÑÛÈÄÅÍÒÈÔÈÊÀÖÈÈ ÓÐÀÂÍÅÍÈÉ

ÒÅÎÐÈÈ ÓÏÐÓÃÎÑÒÈ

Àíèêîíîâ Þ.Å.1, Àþïîâà Í.Á.2, Íåùàäèì Ì.Â.3

Èíñòèòóò ìàòåìàòèêè èì. Ñ.Ë. Ñîáîëåâà ÑÎ ÐÀÍ,Íîâîñèáèðñê, Ðîññèÿ;

[email protected], [email protected], [email protected]

Ïðåäëàãàåòñÿ íîâûé ñïîñîá èññëåäîâàíèÿ çàäà÷ èäåíòèôèêàöèè ñè-ñòåìû óðàâíåíèé óïðóãîñòè, îñíîâàííûé íà ëó÷åâîì ðàçëîæåíèè ðåøå-íèé, êîãäà êîýôôèöèåíòû äàííîé ñèñòåìû çàâèñÿò íå òîëüêî îò ïðî-ñòðàíñòâåííîé ïåðåìåííîé, íî è îò âðåìåíè, ÷òî ïðåäñòàâëÿåò ïðàê-òè÷åñêèé èíòåðåñ. Àëãåáðî-àíàëèòè÷åñêèìè ìåòîäàìè óñòàíàâëèâàþò-ñÿ íîâûå ìíîãî÷èñëåííûå ñâÿçè â êîíå÷íûõ è áåñêîíå÷íûõ âàðèàíòàõìåæäó àìïëèòóäàìè, êîýôôèöèåíòàìè è ôóíêöèÿìè èñòî÷íèêîâ ðàñ-ñìàòðèâàåìûõ äèíàìè÷åñêèõ ñèñòåì òåîðèè óïðóãîñòè.

Ðàáîòà âûïîëíåíà ïðè ôèíàíñîâîé ïîääåðæêå ïðîåêòà ïî ïðîãðàììå Ïðå-

çèäèóìà ÐÀÍ Âû÷èñëèòåëüíàÿ òîìîãðàôèÿ íåîäíîðîäíûõ è àíèçîòðîïíûõ

ñðåä (ïðîåêò 0314-2015-001), ÐÔÔÈ (ïðîåêò 15-01-00745).


1. Àíèêîíîâ Þ.Å., Íåùàäèì Ì.Â. Àëãåáðî-àíàëèòè÷åñêèå ñïîñîáû ïîñòðî-åíèÿ ðåøåíèé äèôôåðåíöèàëüíûõ óðàâíåíèé // Âåñòí. ÍÃÓ. Ñåð. Ìàòå-ìàòèêà, ìåõàíèêà, èíôîðìàòèêà. 2015. Ò. 15, âûï. 2. Ñ. 321.

2. ÀíèêîíîâÞ.Å., Àþïîâà Í.Á.Ëó÷åâûå ðàçëîæåíèÿ è òîæäåñòâà äëÿ óðàâ-íåíèé âòîðîãî ïîðÿäêà ñ ïðèëîæåíèÿìè ê îáðàòíûì çàäà÷àì // Ñèá.æóðí. ÷èñò. è ïðèêë. ìàòåì. (ïðèíÿòà â ïå÷àòü).

3. Íåùàäèì Ì.Â. Ôóíêöèîíàëüíî èíâàðèàíòíûå ðåøåíèÿ ñèñòåìû Ìàêñâåë-ëà // Ñèá. æóðí. èíäóñòð. ìàòåìàòèêè. 2017. Ò. 20, 4. Ñ. 6674.

4. Àíèêîíîâ Þ.Å., Àþïîâà Í.Á., Íåùàäèì Ì.Â. Ëó÷åâîé ìåòîä è âîïðîñûèäåíòèôèêàöèè óðàâíåíèé òåîðèè óïðóãîñòè // Ñèá. æóðí. ÷èñò. è ïðèêë.ìàòåì. (ñäàíà â ïå÷àòü).

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ÎÏÐÅÄÅËÅÍÈÅ ÏÎÂÅÐÕÍÎÑÒÅÉ ÐÀÇÐÛÂÎÂÊÎÝÔÔÈÖÈÅÍÒÎÂ ÓÐÀÂÍÅÍÈß ÏÅÐÅÍÎÑÀ

Áàëàêèíà Å.Þ.


[email protected]

Ðàññìàòðèâàåòñÿ íåñòàöèîíàðíîå ëèíåéíîå äèôôåðåíöèàëüíîå óðàâ-íåíèå

∂f(t, r, ω,E)

∂t+ ω · ∇rf(t, r, ω,E) + µ(t, r, E)f(t, r, ω,E) = J(t, r, ω,E).

Ýòî óðàâíåíèå îïèñûâàåò, â ÷àñòíîñòè, ïðîöåññ ïåðåíîñà ÷àñòèöñêâîçü ñðåäó. Ôóíêöèÿ f(t, r, ω,E) èíòåðïðåòèðóåòñÿ êàê ïëîòíîñòü ïî-òîêà ÷àñòèö â ìîìåíò âðåìåíè t â òî÷êå r ñ ýíåðãèåé E, ëåòÿùèõ âíàïðàâëåíèè ω. Ôóíêöèè µ è J õàðàêòåðèçóþò ñðåäó G, â êîòîðîé ïðî-òåêàåò ïðîöåññ.

Ðàññìàòðèâàåòñÿ çàäà÷à î íàõîæäåíèè ïîâåðõíîñòåé ðàçðûâîâ êîýô-ôèöèåíòîâ óðàâíåíèÿ µ è J . Èíûìè ñëîâàìè, ñòàâèòñÿ âîïðîñ îá îïðå-äåëåíèè âíóòðåííåé ñòðóêòóðû ñðåäû G. Òàêàÿ ïîñòàíîâêà ÿâëÿåòñÿïðîäîëæåíèåì öèêëà èññëåäîâàíèé Ä.Ñ. Àíèêîíîâà [1].

Äëÿ ðåøåíèÿ ïîñòàâëåííîé ïðîáëåìû ñíà÷àëà èññëåäóåòñÿ ïðÿìàÿçàäà÷à î íàõîæäåíèè ïëîòíîñòè ïîòîêà f ïðè çàäàííûõ íà÷àëüíîì óñëî-âèè è ïëîòíîñòè ïàäàþùåãî ïîòîêà (òàêàÿ æå ïîñòàíîâêà, íî â ñëó÷àåíåïðåðûâíûõ êîýôôèöèåíòîâ, áûëà ðàññìîòðåíà À.È. Ïðèëåïêî [2]).Çàòåì óêàçûâàåòñÿ ñïåöèàëüíàÿ ôóíêöèÿ, êîòîðàÿ ïðèíèìàåò íåîãðà-íè÷åííîå çíà÷åíèå òîëüêî íà èñêîìûõ ïîâåðõíîñòÿõ.

Ðàáîòà âûïîëíåíà ïðè ïîääåðæêå Ðîññèéñêîãî ôîíäà ôóíäàìåíòàëüíûõ

èññëåäîâàíèé (ïðîåêò 16-31-00112 ìîë_à).


1. Àíèêîíîâ Ä.Ñ., Êîâòàíþê À.Å., Ïðîõîðîâ È.Â. Èñïîëüçîâàíèå óðàâíåíèÿïåðåíîñà â òîìîãðàôèè. Ì.: Ëîãîñ, 2000.

2. Ïðèëåïêî À.È., Èâàíêîâ À.Ë. Îáðàòíûå çàäà÷è îïðåäåëåíèÿ êîýôôèöè-åíòà è ïðàâîé ÷àñòè íåñòàöèîíàðíîãî ìíîãîñêîðîñòíîãî óðàâíåíèÿ ïåðå-íîñà ïî ïåðåîïðåäåëåíèþ â òî÷êå // Äèôôåðåíö. óðàâíåíèÿ. 1985. Ò. 21, 1. C. 109119.

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ÓÑËÎÂÈß ÀÑÈÌÏÒÎÒÈ×ÅÑÊÎÉÓÑÒÎÉ×ÈÂÎÑÒÈ ÐÅØÅÍÈÉ ÐÀÇÍÎÑÒÍÛÕ

ÓÐÀÂÍÅÍÈÉ Ñ ÏÅÐÈÎÄÈ×ÅÑÊÈÌÈÊÎÝÔÔÈÖÈÅÍÒÀÌÈ Ñ ÇÀÏÀÇÄÛÂÀÍÈÅÌ

Áàëäàíîâ Ä.Ø.

Íîâîñèáèðñêèé ãîñóäàðñòâåííûé óíèâåðñèòåò,Íîâîñèáèðñê, Ðîññèÿ; [email protected]

Â íàñòîÿùåé ðàáîòå ðàññìàòðèâàåòñÿ âîïðîñ îá àñèìïòîòè÷åñêîéóñòîé÷èâîñòè íóëåâîãî ðåøåíèÿ ñèñòåìû ðàçíîñòíûõ óðàâíåíèé

xn+1 = A(n)xn +Bj(n)xn−τ(n), n = 0, 1, . . . , (1)

ãäå

Bj(n) =

B(n) ïðè τ(n) = j,

0 ïðè τ(n) 6= j,

A(n), B(n)N -ïåðèîäè÷åñêèå ìàòðèöû ðàçìåðàm×m. Ìû áóäåì ïðåä-ïîëàãàòü, ÷òî çàïàçäûâàþùèé àðãóìåíò îãðàíè÷åí 1 ≤ τ(n) ≤ τ <∞.

Èìååò ìåñòî ñëåäóþùèé ðåçóëüòàò.Òåîðåìà. Ïðåäïîëîæèì, ÷òî ñóùåñòâóþò ýðìèòîâû ïîëîæèòåëüíî

îïðåäåëåííûå ìàòðèöû H(n), Kj , j = 0, 1, . . . , τ , òàêèå, ÷òî H(0) =H(N), ∆j = Kj−1 −Kj > 0, j = 1, . . . , τ , è ñîñòàâíûå ìàòðèöû

C(n) = −

C00(n) A∗(n)H(n+1)B1(n) ... A∗(n)H(n+1)Bτ (n)

B∗1 (n)H(n+1)A(n) C11(n) ... B∗1 (n)H(n+1)Bτ (n)

......

. . ....

B∗τ (n)H(n+1)A(n) B∗τ (n)H(n+1)B1(n) ... Cττ (n)

,

ãäå

C00(n) = A∗(n)H(n+ 1)A(n)−H(n) +K0,

Cjj(n) = B∗j (n)H(n+ 1)Bj(n)− 1

2∆j , j = 1, . . . , τ − 1,

Cττ (n) = B∗τ (n)H(n+ 1)Bτ (n)−Kτ ,

òàêæå ïîëîæèòåëüíî îïðåäåëåíû ïðè n = 0, . . . , N − 1. Òîãäà íóëåâîå

ðåøåíèå ñèñòåìû (1) àñèìïòîòè÷åñêè óñòîé÷èâî.


1. Äåìèäåíêî Ã.Â., Áàëäàíîâ Ä.Ø. Îá àñèìïòîòè÷åñêîé óñòîé÷èâîñòè ðå-øåíèé ðàçíîñòíûõ óðàâíåíèé ñ çàïàçäûâàíèåì // Âåñòí. ÍÃÓ. Ñåð. Ìà-òåìàòèêà, ìåõàíèêà, èíôîðìàòèêà. 2015. Ò. 15, âûï. 4. Ñ. 5062.

34


ÓÐÀÂÍÅÍÈß ÒÈÏÀ ÓÐÛÑÎÍÀ Â ÇÀÄÀ×ÀÕÂÎÑÑÒÀÍÎÂËÅÍÈß ÒÎ×ÅÊ ÏÎÂÅÐÕÍÎÑÒÈ

Áåëîçóá Â.À., Êîçëîâà Ì. Ã., Ëóêüÿíåíêî Â.À.

Êðûìñêèé ôåäåðàëüíûé óíèâåðñèòåò èì. Â.È. Âåðíàäñêîãî,Ñèìôåðîïîëü, Ðîññèÿ; [email protected]

Çàäà÷à âîññòàíîâëåíèÿ õàðàêòåðíûõ òî÷åê èçîáðàæåíèé ïî äàííûìêîñâåííûõ èçìåðåíèé ÿâëÿåòñÿ âîñòðåáîâàííîé â ðàçëè÷íûõ ïðèëîæå-íèÿõ. Êàê ïðàâèëî, òàêèå çàäà÷è ìîäåëèðóþòñÿ ëèíåéíûìè è íåëè-íåéíûìè îïåðàòîðíûìè óðàâíåíèÿìè ïåðâîãî ðîäà è ÿâëÿþòñÿ íåêîð-ðåêòíî ïîñòàâëåííûìè. Ðàçðàáîòêà óñòîé÷èâûõ ðåãóëÿðèçèðóþùèõ àë-ãîðèòìîâ ñâÿçàíà ñ àäåêâàòíûì èñïîëüçîâàíèåì àïðèîðíîé è äðóãîéèíôîðìàöèè î ðåøåíèè, ìîäåëè, ñïîñîáàõ ïîëó÷åíèÿ äàííûõ êîñâåí-íûõ èçìåðåíèé. Â ðàáîòå ðàññìàòðèâàþòñÿ ìîäåëè â âèäå ðàçíîñòíûõàíàëîãîâ íåëèíåéíûõ èíòåãðàëüíûõ óðàâíåíèé òèïà ñâåðòêè ïåðâîãîðîäà. Ðàçíîîáðàçèå ðåãóëÿðèçèðóþùèõ àëãîðèòìîâ îïðåäåëÿåòñÿ âûáî-ðîì ñïîñîáà ïåðåðàáîòêè äîñòóïíîé èíôîðìàöèè (çíàíèé) â ðåøåíèå [1].Òåì ñàìûì, ôîðìèðóåòñÿ èíòåëëåêòóàëüíàÿ ñèñòåìà, áàçîâûìè ýëåìåí-òàìè êîòîðîé ÿâëÿþòñÿ óëüòðàñèñòåìû (ïî òåðìèíîëîãèè À.Â. ×å÷êè-íà) ïðåîáðàçîâàòåëè ñåìàíòè÷åñêîé èíôîðìàöèè óëüòðàîïåðàòîðû.Íà ïåðâîì ýòàïå ðåøàåòñÿ çàäà÷à âîññòàíîâëåíèÿ ýêñòðåìàëüíûõ òî÷åêïîâåðõíîñòè (áëåñòÿùèõ). Äëÿ ýòîãî èñïîëüçóþòñÿ àñèìïòîòè÷åñêèåìîäåëè [24].

Ñëåäóþùèé øàã ñîñòîèò â ñèíòåçå èíòåëëåêòóàëüíîé ñèñòåìû ïî îá-ðàáîòêå äàííûõ êîñâåííûõ èçìåðåíèé èç ìíîæåñòâà âçàèìîäåéñòâóþ-ùèõ èíòåëëåêòóàëüíûõ àãåíòîâ (ÈÀ) è ñîîòâåòñòâóþùåé èíòåëëåêòó-àëüíîé ñèñòåìû óïðàâëåíèÿ (ÈÑÓ). Èíôîðìàöèÿ, ïîëó÷åííàÿ îäíèìÈÀ, ìîæåò áûòü ïðåöåäåíòíîé, àïðèîðíîé äëÿ äðóãîãî ÈÀ. Íàïðè-ìåð, äâà ÈÀ ðåøàþò çàäà÷ó äèñòàíöèîííîãî çîíäèðîâàíèÿ âûäåëåí-íîãî ó÷àñòêà ïîâåðõíîñòè (òðàññû). Ðåøåíèå, ïîëó÷åííîå â âèäå íàáîðàýêñòðåìàëüíûõ òî÷åê îäíèì ÈÀ, ìîæåò èòåðàöèîííî óòî÷íÿòüñÿ äðó-ãèì. ÈÀ ðàçìåùàþòñÿ íà îäíîì óñòðîéñòâå èëè íåñêîëüêèõ. Ñèñòåìûóðàâíåíèé ìîäåëèðóþò ïðîöåññ ñêàíèðîâàíèÿ ïîâåðõíîñòè àíòåííûìèóñòðîéñòâàìè, õàðàêòåð ñèãíàëà è åãî îòðàæåíèå îò ñêàíèðóåìîé ïî-âåðõíîñòè. Òàêèå íåëèíåéíûå ñèñòåìû àëãåáðàè÷åñêèõ óðàâíåíèé ÿâ-ëÿþòñÿ àíàëîãàìè íåëèíåéíûõ èíòåãðîäèôôåðåíöèàëüíûõ óðàâíåíèéòèïà Óðûñîíà 1 ðîäà.

35



1. Âàñèí Â.Â., Åðåìèí È.È. Îïåðàòîðû è èòåðàöèîííûå ïðîöåññû ôåéåðîâ-ñêîãî òèïà (òåîðèÿ è ïðèëîæåíèÿ). Ìîñêâà Èæåâñê: Èíñòèòóò êîìïüþ-òåðíûõ èññëåäîâàíèé, ÍÈÖ Ðåãóëÿðíàÿ è õàîòè÷åñêàÿ äèíàìèêà, 2005.

2. Ëóêüÿíåíêî Â.À., Õàçîâà Þ.À. Ðåøåíèå íåëèíåéíûõ èíòåãðàëüíûõ óðàâ-íåíèé Óðûñîíà òèïà ñâåðòêè // Àíàëèç, ìîäåëèðîâàíèå, óïðàâëåíèå, ðàç-âèòèå ýêîíîìè÷åñêèõ ñèñòåì: ñáîðíèê íàó÷íûõ òðóäîâ V Ìåæäóíàðîäíîéøêîëû-ñèìïîçèóìà ÀÌÓÐ-2011. Ñèìôåðîïîëü: ÒÍÓ èì. Â.È. Âåðíàäñêî-ãî, 2011. Ñ. 214220.

3. Ëóêüÿíåíêî Â.À., Áåëîçóá Â.À. Íåëèíåéíûå óðàâíåíèÿ òèïà ñâåðòêè ñäåëüòîîáðàçíûìè ÿäðàìè // XXVI Êðûìñêàÿ Îñåííÿÿ Ìàòåìàòè÷åñêàÿØêîëà-ñèìïîçèóì ïî ñïåêòðàëüíûì è ýâîëþöèîííûì çàäà÷àì. Ñá. òåç.äîêë. Ñèìôåðîïîëü: ÎÎÎ ÔÎÐÌÀ, 2015. Ñ. 95.

4. Lukyanenko V.A. Some tasks for integral equations of Urison's type //Proceedings of the International Conference Integral Equations2010. Lviv,Ukraine: PAIS, 2010. Ð. 8084.

36


ÊÐÀÅÂÀß ÇÀÄÀ×À ÎÁ ÎÑÅÑÈÌÌÅÒÐÈ×ÍÎÌÎÁÒÅÊÀÍÈÈ ÊÐÓÃÎÂÎÃÎ ÊÎÍÓÑÀ

ÐÅÀËÜÍÛÌ ÃÀÇÎÌ

Áèáåðäîðô Ý.À.1, Áëîõèí À.Ì.2

Íîâîñèáèðñêèé ãîñóäàðñòâåííûé óíèâåðñèòåò,Èíñòèòóò ìàòåìàòèêè èì. Ñ.Ë. Ñîáîëåâà ÑÎ ÐÀÍ,

Íîâîñèáèðñê, Ðîññèÿ; [email protected], [email protected]

Ðàññìàòðèâàåòñÿ çàäà÷à î ñòàöèîíàðíîì îáòåêàíèè êðóãëîãî êîíó-ñà ãàçîì Âàí-äåð-Âààëüñà [1], îïèñûâàþùèì âåùåñòâî â ãàçîîáðàçíîì,æèäêîì, à òàêæå äâóõôàçíîì ñîñòîÿíèè. Àêòóàëüíîñòü äàííîé çàäà÷èîáóñëîâëåíà ñîâðåìåííûì óðîâíåì ðàçâèòèÿ ëåòàòåëüíûõ àïïàðàòîâ,äâèæóùèõñÿ ñî ñâåðõ- è ãèïåðçâóêîâûìè ñêîðîñòÿìè.

Äàííàÿ çàäà÷à îïèñûâàåòñÿ ñèñòåìîé ÎÄÓ ÷åòâåðòîãî ïîðÿäêà, îïðå-äåëåííîé íà èíòåðâàëå, îäèí êîíåö êîòîðîãî (óãîë óäàðíîé âîëíû) òàê-æå ÿâëÿåòñÿ íåèçâåñòíîé âåëè÷èíîé, è ïÿòüþ ãðàíè÷íûìè óñëîâèÿìè [2].Ðàçðàáîòàííûé àëãîðèòì íà êàæäîé èòåðàöèè ðåøàåò ñòàíäàðòíóþ êðà-åâóþ çàäà÷ó, èñïîëüçóÿ ÷åòûðå ãðàíè÷íûõ óñëîâèÿ, à çàòåì ïðîèçâîäèòêîððåêòèðîâêó ðåøåíèÿ íà îñíîâå ïÿòîãî ãðàíè÷íîãî óñëîâèÿ. Àëãî-ðèòì áûñòðî ñõîäèòñÿ (23 èòåðàöèè) äëÿ äîñòàòî÷íî áîëüøèõ ÷èñåëÌàõà íàáåãàþùåãî ïîòîêà.

Âàðèàöèÿ àëãîðèòìà, âû÷èñëÿþùàÿ ÷èñëî Ìàõà íàáåãàþùåãî ïîòî-êà, ñîîòâåòñòâóþùåå çàäàííîìó óãëó óäàðíîé âîëíû, ïîçâîëÿåò îïðåäå-ëèòü ìèíèìàëüíîå ÷èñëî Ìàõà, ïðè êîòîðîì ñóùåñòâóåò ïðèñîåäèíåí-íàÿ óäàðíàÿ âîëíà.

Ïðîèçâåäåíî ñðàâíåíèå ðàñ÷åòíûõ ïàðàìåòðîâ òå÷åíèé ãàçà Âàí-äåð-Âààëüñà ñ òå÷åíèÿìè ïîëèòðîïíîãî ãàçà.




1. Áëîõèí À.Ì., Ãîëäèí À.Þ. Ïîñòðîåíèå ïðîìåæóòî÷íûõ îáëàñòåé äëÿîáîáùåííîãî ãàçà Âàí-äåð-Âààëüñà //Æóðí. òåõ. ôèç. 2016. Ò. 86, âûï. 12.Ñ. 4955.

2. Áëîõèí À.Ì., Áèáåðäîðô Ý.À. ×èñëåííîå ðåøåíèå çàäà÷è î ñòàöèîíàðíîìîáòåêàíèè êîíóñà ðåàëüíûì ãàçîì // Âû÷èñëèòåëüíûå òåõíîëîãèè. 2015.Ò. 20, 2. Ñ. 2943.

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ÒÐÀÅÊÒÎÐÍÛÉ È ÃËÎÁÀËÜÍÛÉÀÒÒÐÀÊÒÎÐÛ ÎÄÍÎÃÎ ÊËÀÑÑÀ

ÂßÇÊÎÓÏÐÓÃÈÕ ÆÈÄÊÎÑÒÅÉ Ñ ÏÀÌßÒÜÞ

Áîëäûðåâ À.Ñ.

Âîðîíåæñêèé ãîñóäàðñòâåííûé óíèâåðñèòåò, Âîðîíåæ, Ðîññèÿ;[email protected]

Ïóñòü Ω ⊂ Rn îãðàíè÷åííàÿ îáëàñòü ñ ëîêàëüíî-ëèïøèöåâîé ãðà-íèöåé Γ. Ðàññìîòðèì ñëåäóþùóþ ñèñòåìó óðàâíåíèé

∂v

∂t+

n∑i=1

vi∂v

∂xi− µ1 Div

∫ t

0

L(t, s)E(v)(s, Zδ(v)(s; t, x)) ds

−µ0 Div E(v) = −grad p+ f, (t, x) ∈ [0,+∞)× Ω, (1)

div v = 0, v |Γ = 0, v(0, x) = v0(x),

∫Ω

p dx = 0. (2)

Çäåñü v ñêîðîñòü, p äàâëåíèå, f ïëîòíîñòü âíåøíèõ ñèë.E = (Eij) òåíçîð ñêîðîñòåé äåôîðìàöèè, µ0 = 2κ/λ, µ1 = 2ν/λ −2κ/λ2, ãäå λ âðåìÿ ðåëàêñàöèè, κ âðåìÿ çàïàçäûâàíèÿ, ν âÿç-êîñòü æèäêîñòè. L(t, s) èçìåðèìàÿ ôóíêöèÿ, õàðàêòåðèçóþùàÿ ïà-ìÿòü ÷àñòèöû æèäêîñòè. Ïðåäïîëàãàåòñÿ, ÷òî |L(t, s)| ≤ e−(2µ1/µ0)(t−s)

(s < t, s, t ∈ [0,+∞)). Ðàññìîòðèì òðàåêòîðèþ, îïðåäåëÿåìóþ óðàâíå-íèåì z(τ ; t, x) = x+

∫ τtSδv(s, z(s; t, x))ds. Â ýòîì óðàâíåíèè èñïîëüçóåòñÿ

îïåðàòîð ðåãóëÿðèçàöèè Sδ òàêîé, ÷òî äëÿ êàæäîãî v ∈ L2(0, T ;V ) ýòîóðàâíåíèå èìååò åäèíñòâåííîå ðåøåíèå Zδ(v). Ðàçðåøèìîñòü â ñëàáîìñìûñëå çàäà÷è (1)(2) óñòàíîâëåíà â ðàáîòå [1]. Ïîäîáíûå ìîäåëè ðàñ-ñìàòðèâàþòñÿ, íàïðèìåð, â [2].

Òåîðåìà. Ïóñòü f ∈ V ∗. Òîãäà ñóùåñòâóåò ìèíèìàëüíûé òðàåê-

òîðíûé àòòðàêòîð U è ãëîáàëüíûé àòòðàêòîð A ïðîñòðàíñòâà òðàåêòî-

ðèé H+ çàäà÷è (1)(2).


1. Áîëäûðåâ À.Ñ. Èññëåäîâàíèå îäíîãî êëàññà âÿçêîóïðóãèõ æèäêîñòåé ñïàìÿòüþ // Âåñòí. Âîðîíåæ. ãîñ. óí-òà. Ñåð. Ôèçèêà, ìàòåìàòèêà. 2015. 4. Ñ. 6277.

2. Çâÿãèí Â. Ã., Äìèòðèåíêî Â.Ò. Î ñëàáûõ ðåøåíèÿõ íà÷àëüíî-êðàåâîé çà-äà÷è äëÿ óðàâíåíèé äâèæåíèÿ âÿçêîóïðóãîé æèäêîñòè // ÄÀÍ. 2001.Ò. 380, 3. Ñ. 308311.

38


O ÐÀÇÐÅØÈÌÎÑÒÈ ÂÒÎÐÎÉ ÊÐÀÅÂÎÉÇÀÄÀ×È ÄËß ÑÈÑÒÅÌÛ ÑÒÎÊÑÀ

Áîíäàðü Ë.Í.


[email protected]

Ðàññìàòðèâàåòñÿ âòîðàÿ êðàåâàÿ çàäà÷à â ïîëóïðîñòðàíñòâå R3+ =

x = (x′, x3) : x′ ∈ R2, x3 > 0 äëÿ ñèñòåìû Ñòîêñà:−ν4u+ γu+∇p = f+(x), x ∈ R3

+,div u = f−(x),−σi3p+ ν

(Dx3u

i +Dxiu3) ∣∣x3=0

= 0, i = 1, 2, 3,

ãäå u = (u1, u2, u3)T , f = (f+, f−)T = (f1, f2, f3, f−)T , ν, γ > 0, σi3 ñèìâîë Êðîíåêåðà. Èññëåäóåòñÿ ðàçðåøèìîñòü êðàåâîé çàäà÷è â ñîáî-ëåâñêîì ïðîñòðàíñòâå W 2

q (R3+)×W 1

q (R3+).

Òåîðåìà. Ïóñòü q > 3/2. Òîãäà äëÿ f+(x) ∈ Lq(R3+) ∩ L1(R3

+),f−(x) ∈ W 1

q (R3+), (1 + |x|)f−(x) ∈ L1(R3

+), f−(x′, 0) ≡ 0 ñóùåñòâóåò

åäèíñòâåííîå ðåøåíèå (u(x), p(x))T ∈ W 2q (R3

+)×W 1q (R3

+) êðàåâîé çàäà-

÷è.

Çàìå÷àíèå 1. Èç ðàáîòû [1] âûòåêàåò, ÷òî ïðè f+(x) ∈ Lq(R3+),

f−(x) ≡ 0 êðàåâàÿ çàäà÷à èìååò åäèíñòâåííîå ðåøåíèå u(x) ∈ W 2q (R3

+),∇p(x) ∈ Lq(R3

+), 1 < q <∞.Çàìå÷àíèå 2. Óñëîâèå q > 3/2, óêàçàííîå â òåîðåìå, ÿâëÿåòñÿ ñó-

ùåñòâåííûì. Ìîæíî ïîêàçàòü, ÷òî ïðè 1 < q ≤ 3/2 âîçíèêàþò äîïîë-íèòåëüíûå óñëîâèÿ íà ïðàâóþ ÷àñòü ñèñòåìû f(x). Àíàëîãè÷íûé ôàêòèìååò ìåñòî è äëÿ çàäà÷è Êîøè è ïåðâîé êðàåâîé çàäà÷è äëÿ ëèíåàðè-çîâàííîé ñèñòåìû Íàâüå Ñòîêñà (ñì. [2, 3]).


1. Ñîëîííèêîâ Â.À. Îöåíêè ðåøåíèÿ îäíîé íà÷àëüíî-êðàåâîé çàäà÷è äëÿ ëè-íåéíîé íåñòàöèîíàðíîé ñèñòåìû óðàâíåíèé Íàâüå Ñòîêñà // Çàï. íàó÷í.ñåì. ËÎÌÈ. 1976. Ò. 59. C. 178254.

2. Äåìèäåíêî Ã.Â. Î íåîáõîäèìûõ óñëîâèÿõ êîððåêòíîñòè çàäà÷è Êîøè äëÿëèíåàðèçîâàííîé ñèñòåìû Íàâüå Ñòîêñà // Ñèá. ìàò. æóðí. 1988. Ò. 29, 3. Ñ. 186189.

3. Àëåêñàíäðîâñêèé È.Ë. Óñëîâèå ðàçðåøèìîñòè ïåðâîé êðàåâîé çàäà÷è äëÿëèíåàðèçîâàííîé ñèñòåìû Íàâüå Ñòîêñà // Ñèá. ìàò. æóðí. 1998. Ò. 39, 6. Ñ. 12111224.

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ÃÐÓÏÏÎÂÎÉ ÀÍÀËÈÇ ÎÄÍÎÌÅÐÍÎÃÎÓÐÀÂÍÅÍÈß ÁÎËÜÖÌÀÍÀ

Áîðîâñêèõ À.Â.1, Ïëàòîíîâà K.C.2

Ìîñêîâñêèé ãîñóäàðñòâåííûé óíèâåðñèòåò èì. Ì.Â. Ëîìîíîñîâà,Ìîñêâà, Ðîññèÿ; [email protected], [email protected]

Ïðîâåäåíà ãðóïïîâàÿ êëàññèôèêàöèÿ îäíîìåðíîãî óðàâíåíèÿ Áîëüö-ìàíà ft + cfx + F(t, x, c)fc = 0 îòíîñèòåëüíî ôóíêöèè F = F(t, x, c),õàðàêòåðèçóþùåé âíåøíåå ñèëîâîå ïîëå, â ïðåäïîëîæåíèè íàëè÷èÿ äî-ïîëíèòåëüíûõ ñîîòíîøåíèé ìåæäó ïåðåìåííûìè, îïðåäåëÿåìûõ ôèçè-÷åñêèì ñìûñëîì èñïîëüçóåìûõ âåëè÷èí. Ïîêàçàíî, ÷òî äëÿ âñåõ F àë-ãåáðà ñèììåòðèé êîíå÷íîìåðíà è ìîæåò èìåòü ðàçìåðíîñòè 1, 2, 3, 8.Ìàêñèìàëüíàÿ ðàçìåðíîñòü ðåàëèçóåòñÿ ïðè F ≡ 0.

Òåîðåìà.1. Àëãåáðà ñèììåòðèé óðàâíåíèÿ ft + cfx + F(t, x, c)fc = 0 (t, x, c ∈ R),ñîõðàíÿþùàÿ óðàâíåíèÿ õàðàêòåðèñòèê dx = cdt, dc = Fdt è ïðÿìûå

dt = dx = 0, êîíå÷íîìåðíà äëÿ ëþáîé ôóíêöèè F(t, x, c).2. Êîíå÷íîìåðíûå íåòðèâèàëüíûå àëãåáðû ñèììåòðèé èìåþò óðàâíåíèÿ

ñ ôóíêöèÿìè F(t, x, c) êëàññîâ, ïðåäñòàâèòåëè êîòîðûõ ïåðå÷èñëåíû íè-

æå. Êàæäûé êëàññ ïîëó÷àåòñÿ êàê îðáèòà ãðóïïû äèôôåîìîðôèçìîâ â

ïðîñòðàíñòâå ïåðåìåííûõ (t, x). Â êàæäîì ñëó÷àå IIV ïðåäïîëàãàåòñÿ,

÷òî ôóíêöèÿ F íå ëåæèò â ïðåäûäóùåì êëàññå.

I. F = 0, ãðóïïà ñèììåòðèé âîñüìèìåðíà;

II. F = Aca, F = A exp(ac), F = A exp

∫3c+ a

c2 + bc+ ddc, F =

A

x3,

F = A

(1 +

(t+ ac)2

t2 + 2ax

)3/2

, F = A

((x− ct)2 + c2 + 1

x2 + t2 + 1

)3/2

, ãðóïïà ñèì-

ìåòðèé òð¼õìåðíà;

III. F = F(c), F = T (c)/t, ãðóïïà ñèììåòðèé äâóìåðíà;

IV. F = F(x, c), ãðóïïà ñèììåòðèé îäíîìåðíà.

Äëÿ êàæäîãî ñëó÷àÿ îïðåäåëåíû àëãåáðû ñèììåòðèé ñîîòâåòñòâóþ-ùåé ãðóïïû.



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ÑÕÎÄÈÌÎÑÒÜ ÊÓÁÀÒÓÐÍÛÕ ÔÎÐÌÓËÂÛÑÎÊÎÉ ÒÐÈÃÎÍÎÌÅÒÐÈ×ÅÑÊÎÉ

ÒÎ×ÍÎÑÒÈ

Âàñêåâè÷ Â.Ë.


Â äîêëàäå äàíà îöåíêà ñâåðõó ïîðÿäêà ñõîäèìîñòè íîðì ôóíêöè-îíàëîâ ïîãðåøíîñòè êóáàòóðíûõ ôîðìóë âûñîêîé òðèãîíîìåòðè÷åñêîéòî÷íîñòè ïðè óâåëè÷åíèè ÷èñëà óçëîâ. Äåéñòâèå ôóíêöèîíàëîâ ïîãðåø-íîñòè ðàññìàòðèâàåòñÿ íà ìíîãîìåðíûõ ïåðèîäè÷åñêèõ ïðîñòðàíñòâàõÑîáîëåâà, âêëþ÷àÿ ïðîñòðàíñòâà äðîáíîé ãëàäêîñòè. Îñíîâíîé ðåçóëü-òàò ïîëó÷åí ïðè âûïîëíåíèè îáùåïðèíÿòûõ óñëîâèé íà ãëàäêîñòü ïðî-ñòðàíñòâà ïîäûíòåãðàëüíûõ ôóíêöèé è íà ðàñïðåäåëåíèå óçëîâ êóáà-òóðíûõ ôîðìóë [13].

Ïðè îöåíêå ïîðÿäêà ñõîäèìîñòè èñïîëüçîâàíû íàéäåííûå ÿâíûåîöåíêè ñâåðõó â âèäå ñòåïåííîé ôóíêöèè äëÿ çíà÷åíèé êðàòíîñòè rn(p)ñîáñòâåííûõ çíà÷åíèé îïåðàòîðà Ëàïëàñà â ñëó÷àå ïåðèîäè÷åñêèõ óñëî-âèé [4].


1. Ñîáîëåâ Ñ.Ë., Âàñêåâè÷ Â.Ë. Êóáàòóðíûå ôîðìóëû. Íîâîñèáèðñê: Èçä-âîÈíñòèòóòà ìàòåìàòèêè, 1996.

2. Âàñêåâè÷ Â.Ë. Ïîãðåøíîñòü è ãàðàíòèðîâàííàÿ òî÷íîñòü êóáàòóðíûõôîðìóë â ìíîãîìåðíûõ ïåðèîäè÷åñêèõ ïðîñòðàíñòâàõ Ñîáîëåâà // Ñèá.ìàò. æóðí. 2014. Ò. 55, 5. Ñ. 971988.

3. Âàñêåâè÷ Â.Ë. Ñõîäèìîñòü êóáàòóðíûõ ôîðìóë âûñîêîé òðèãîíîìåòðè-÷åñêîé òî÷íîñòè â ìíîãîìåðíûõ ïåðèîäè÷åñêèõ ïðîñòðàíñòâàõ Ñîáîëå-âà // Ìàò. òðóäû. 2015. Ò. 18, 1. Ñ. 314.

4. Âàñêåâè÷ Â.Ë.Ìàæîðàíòà êðàòíîñòè ñîáñòâåííûõ ÷èñåë ëàïëàñèàíà ñ ïå-ðèîäè÷åñêèìè óñëîâèÿìè // Ìàò. òðóäû. 2017. Ò. 20, 1. Ñ. 7580.

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ÀËÃÅÁÐÀÈ×ÅÑÊÈÅ ÏÅÐÂÛÅ ÈÍÒÅÃÐÀËÛÏÎËÈÍÎÌÈÀËÜÍÛÕ ÑÈÑÒÅÌ,

ÓÄÎÂËÅÒÂÎÐßÞÙÈÕ ÓÑËÎÂÈßÌÊÎØÈ ÐÈÌÀÍÀ

Âîëîêèòèí Å.Ï.


Ìû ðàññìàòðèâàåì ïëîñêóþ ïîëèíîìèàëüíóþ ñèñòåìó äèôôåðåíöè-àëüíûõ óðàâíåíèé

x = P (x, y), y = Q(x, y), (PCR)

ãäå P (x, y), Q(x, y) ìíîãî÷ëåíû ñòåïåíè n îò äâóõ ïåðåìåííûõ x, yñ äåéñòâèòåëüíûìè êîýôôèöèåíòàìè, óäîâëåòâîðÿþùèå óðàâíåíèÿìÊîøè Ðèìàíà

Px(x, y) = Qy(x, y), Py(x, y) = −Qx(x, y).

Òåîðåìà 1. Åñëè n ≥ 2, òî ïîëèíîìèàëüíàÿ ñèñòåìà (PCR) ñòåïåíèn íå èìååò ïîëèíîìèàëüíîãî ïåðâîãî èíòåãðàëà.

Òåîðåìà 2. Ïóñòü âñå òî÷êè ïîêîÿ ñèñòåìû (PCR) ÿâëÿþòñÿ èëè

öåíòðàìè, èëè äèêðèòè÷åñêèìè óçëàìè, è ñîáñòâåííûå ÷èñëà ðàöèî-

íàëüíî ñîèçìåðèìû. Òîãäà ñèñòåìà (PCR) èìååò ðàöèîíàëüíûé ïåðâûéèíòåãðàë.

Òåîðåìà 3. Îäíîðîäíàÿ ñèñòåìà (PCR) èìååò ðàöèîíàëüíûé ïåð-

âûé èíòåãðàë.

Íàøà ãèïîòåçà ñîñòîèò â òîì, ÷òî îïèñàííûå â òåîðåìàõ 2, 3 êëàñ-ñû èñ÷åðïûâàþò ñïèñîê ðàöèîíàëüíî èíòåãðèðóåìûõ ïîëèíîìèàëüíûõñèñòåì âèäà (PCR).




1. Volokitin E. P. Algebraic rst integrals of the polynomial systems satisfying theCauchyRiemann conditions // Qual. Theory Dyn. Syst. 2016. V. 15, No 2.P. 575596.

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Î ÒÎÏÎËÎÃÈÈ (q1, q2)-ÊÂÀÇÈÌÅÒÐÈ×ÅÑÊÈÕÏÐÎÑÒÐÀÍÑÒÂ

Ãðåøíîâ À.Â.


[email protected]

Ïàðà (X, d), ãäå X íåêîòîðîå ìíîæåñòâî, ñîñòîÿùåå íå ìåíåå, ÷åìèç äâóõ ýëåìåíòîâ, d íåîòðèöàòåëüíàÿ ôóíêöèÿ, îïðåäåëåííàÿ íàX × X, íàçûâàåòñÿ (q1, q2)-êâàçèìåòðè÷åñêèì ïðîñòðàíñòâîì [1], åñ-ëè âûïîëíÿåòñÿ

d(x, y) = 0 ⇔ x = y ∀x, y ∈ X,d(x, z) ≤ q1d(x, y) + q2d(y, z) ∀x, y, z ∈ X,

ãäå q1, q2 íåêîòîðûå ïîëîæèòåëüíûå êîíñòàíòû. Ê (q1, q2)-êâàçèìåò-ðè÷åñêèì ïðîñòðàíñòâàì îòíîñÿòñÿ, â ÷àñòíîñòè, ýêâèðåãóëÿðíûå ïðî-ñòðàíñòâà Êàðíî Êàðàòåîäîðè ñ Box-êâàçèìåòðèêàìè è èõ îáîáùå-íèÿ [2].

Áóäåì ãîâîðèòü, ÷òî (q1, q2)-êâàçèìåòðè÷åñêîå ïðîñòðàíñòâî óäîâëå-òâîðÿåò óñëîâèþ lim-ñëàáîé ñèììåòðèè, åñëè

limn→∞

d(x0, xn) = 0 ⇒ limn→∞d(xn, x0) = 0.

Íàìè ïîëó÷åíû ðåçóëüòàòû î âûïîëíåíèè àêñèîì îòäåëèìîñòè äëÿ(q1, q2)-êâàçèìåòðè÷åñêèõ ïðîñòðàíñòâ, óäîâëåòâîðÿþùèõ óñëîâèþ lim-ñëàáîé ñèììåòðèè.




1. Àðóòþíîâ À.Â., Ãðåøíîâ À.Â. Òåîðèÿ (q1, q2)-êâàçèìåòðè÷åñêèõ ïðî-ñòðàíñòâ è òî÷êè ñîâïàäåíèÿ // ÄÀÍ. 2016. Ò. 469, 5. Ñ. 527531.

2. Ãðåøíîâ À.Â. Äîêàçàòåëüñòâî òåîðåìû Ãðîìîâà îá îäíîðîäíîé íèëüïî-òåíòíîé àïïðîêñèìàöèè äëÿ âåêòîðíûõ ïîëåé êëàññà C1 // Ìàò. òðóäû.2012. Ò. 15, 2. Ñ. 7288.

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ÎÁ ÓÑÒÎÉ×ÈÂÎÑÒÈ ÐÅØÅÍÈÉÎÄÍÎÃÎ ÊËÀÑÑÀ ÑÈÑÒÅÌ

ÍÅÉÒÐÀËÜÍÎÃÎ ÒÈÏÀ Ñ ÏÀÐÀÌÅÒÐÎÌ

Äâîðíèöêèé Â.ß.

Íîâîñèáèðñêèé ãîñóäàðñòâåííûé óíèâåðñèòåò, Íîâîñèáèðñê, Ðîññèÿ;[email protected]

Â ðàáîòå ðàññìàòðèâàþòñÿ ñèñòåìû äèôôåðåíöèàëüíûõ óðàâíåíèéñ çàïàçäûâàþùèì àðãóìåíòîì ñëåäóþùåãî âèäà

d

dty(t) = [A0 +A1(ωt)]y(t) + C

d

dty(t− τ), t ≥ 0, (1)

ãäå A0 ïîñòîÿííàÿ ìàòðèöà, ñïåêòð êîòîðîé ëåæèò â ëåâîé ïîëóïëîñ-

êîñòè, A1(t) T -ïåðèîäè÷åñêàÿ ìàòðèöà, ïðè÷åì

T∫0

A1(t) dt = 0, C

ïîñòîÿííàÿ ìàòðèöà, τ > 0 ïàðàìåòð çàïàçäûâàíèÿ, ω > 0 ïàðà-ìåòð.

Åñëè C = 0, êàê ñëåäóåò èç [1], ñóùåñòâóåò ÷èñëî ω0 > 0 òàêîå,÷òî ïðè âñåõ ω ≥ ω0 íóëåâîå ðåøåíèå ñèñòåìû îáûêíîâåííûõ äèôôå-ðåíöèàëüíûõ óðàâíåíèé (1) àñèìïòîòè÷åñêè óñòîé÷èâî. Ïðè C 6= 0 ìûóêàçûâàåì óñëîâèå íà íîðìó ìàòðèöû C è çíà÷åíèå ω1 > 0 òàêîå, ÷òîïðè âñåõ ω ≥ ω1 íóëåâîå ðåøåíèå ñèñòåìû íåéòðàëüíîãî òèïà (1) ýêñïî-íåíöèàëüíî óñòîé÷èâî. Ïðè ïðîâåäåíèè ðàññóæäåíèé ìû îïèðàåìñÿ íàðåçóëüòàòû èç [2].

Àâòîð âûðàæàåò áëàãîäàðíîñòü ê.ô.-ì.í. È.È. Ìàòâååâîé çà ïîñòà-íîâêó çàäà÷è è íàó÷íîå ðóêîâîäñòâî.




1. Äàëåöêèé Þ.Ë., Êðåéí Ì.Ã. Óñòîé÷èâîñòü ðåøåíèé äèôôåðåíöèàëüíûõóðàâíåíèé â áàíàõîâîì ïðîñòðàíñòâå. Ì.: Íàóêà, 1970.

2. Ìàòâååâà È.È. Îá ýêñïîíåíöèàëüíîé óñòîé÷èâîñòè ðåøåíèé ïåðèîäè÷å-ñêèõ ñèñòåì íåéòðàëüíîãî òèïà // Ñèá. ìàò. æóðí. 2017. Ò. 58, 2. Ñ. 344352.

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Î ÇÀÄÀ×Å ÊÎØÈ ÄËßÏÑÅÂÄÎÃÈÏÅÐÁÎËÈ×ÅÑÊÈÕ ÓÐÀÂÍÅÍÈÉ

Äåìèäåíêî Ã.Â.


[email protected]

Â ìîíîãðàôèè [1] áûëà ââåäåíà íåêîòîðàÿ êëàññèôèêàöèÿ óðàâíå-íèé, íå ðàçðåøåííûõ îòíîñèòåëüíî ñòàðøåé ïðîèçâîäíîé

L0(Dx)Dltu+

l−1∑k=0

Ll−k(Dx)Dkt u = f(t, x), t > 0, x ∈ Rn.

Áûëî âûäåëåíî òðè êëàññà óðàâíåíèé: óðàâíåíèÿ ñîáîëåâñêîãî òèïà,ïñåâäîïàðàáîëè÷åñêèå è ïñåâäîãèïåðáîëè÷åñêèå óðàâíåíèÿ. Äëÿ ïåðâûõäâóõ êëàññîâ â [1] áûëà ïîñòðîåíà Lp-òåîðèÿ êðàåâûõ çàäà÷. Äëÿ ïñåâäî-ãèïåðáîëè÷åñêèõ óðàâíåíèé áåç ìëàäøèõ ÷ëåíîâ (â îáîáùåííîì ñìûñ-ëå) â [1] áûëè âïåðâûå ïîëó÷åíû ýíåðãåòè÷åñêèå îöåíêè è äîêàçàíûòåîðåìû î ðàçðåøèìîñòè â ñîáîëåâñêèõ ïðîñòðàíñòâàõ. Îòìåòèì, ÷òî âýòîò êëàññ âõîäÿò, â ÷àñòíîñòè, ëèíåàðèçîâàííûå óðàâíåíèÿ, âîçíèêà-þùèå ïðè èçó÷åíèè êîëåáàíèé ñòåðæíåé, áàëîê è äð.: ìîäåëè Ðýëåÿ Áèøîïà, Ðýëåÿ Ëÿâà è ò.ä. (ñì., íàïðèìåð, [2] è èìåþùóþñÿ áèáëèî-ãðàôèþ). Îáîáùåíèå ðåçóëüòàòîâ [1] ñîäåðæèòñÿ â [3, 4]. Â íàñòîÿùåéðàáîòå ìû ïðîäîëæàåì ýòè èññëåäîâàíèÿ.


1. Äåìèäåíêî Ã.Â., Óñïåíñêèé Ñ.Â. Óðàâíåíèÿ è ñèñòåìû, íå ðàçðåøåííûåîòíîñèòåëüíî ñòàðøåé ïðîèçâîäíîé. Íîâîñèáèðñê: Íàó÷íàÿ êíèãà, 1998.

2. Ãåðàñèìîâ Ñ.È., Åðîôååâ Â.È. Çàäà÷è âîëíîâîé äèíàìèêè ýëåìåíòîâ êîí-ñòðóêöèé. Ñàðîâ: ÔÃÓÏ ÐÔßÖ-ÂÍÈÈÝÔ, 2014.

3. Fedotov I., Volevich L.R. The Cauchy problem for hyperbolic equations notresolved with respect to the highest time derivative // Russian J. Math.Physics. 2006. V. 13, No 3. P. 278292.

4. Äåìèäåíêî Ã.Â. Óñëîâèÿ ðàçðåøèìîñòè çàäà÷è Êîøè äëÿ ïñåâäîãèïåðáî-ëè÷åñêèõ óðàâíåíèé // Ñèá. ìàò. æóðí. 2015. Ò. 56, 6. Ñ. 12891303.

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ÎÁ ÎÄÍÎÌ ÌÅÒÎÄÅ ÈÇÓ×ÅÍÈßÊÀ×ÅÑÒÂÅÍÍÛÕ ÑÂÎÉÑÒÂ ÐÅØÅÍÈÉÏÅÐÂÎÉ ÍÀ×ÀËÜÍÎ-ÊÐÀÅÂÎÉ ÇÀÄÀ×ÈÄËß ÓÐÀÂÍÅÍÈÉ ÑÎÁÎËÅÂÑÊÎÃÎ ÒÈÏÀ

Äåíèñîâà Ò. Å.

Ìîñêîâñêèé ãîðîäñêîé ïñèõîëîãî-ïåäàãîãè÷åñêèé óíèâåðñèòåò,Ìîñêâà, Ðîññèÿ; [email protected], [email protected]

Ïðîäîëæàåòñÿ èçó÷åíèå êà÷åñòâåííûõ ñâîéñòâ (âûõîä íà ïîëèíîì,îñöèëëÿöèÿ, ïî÷òè ïåðèîäè÷íîñòü, ìîíîòîííûé ðîñò, óáûâàíèå ê íó-ëþ (ñ îïðåäåë¼ííîé ñêîðîñòüþ) ñ ðîñòîì âðåìåíè) ðåøåíèé óðàâíåíèéñîáîëåâñêîãî òèïà âèäà

D2t∇(A(x)∇u) +∇(B(x)∇u) = 0

ñ ïîñòîÿííûìè è ïåðåìåííûìè êîýôôèöèåíòàìè; A(x) è B(x) ñîîò-âåòñòâóþùèå ìàòðèöû êîýôôèöèåíòîâ.

Ïðîáëåìàòèêå óðàâíåíèé, íå ðàçðåø¼ííûõ îòíîñèòåëüíî ñòàðøåéïðîèçâîäíîé, ïîñâÿùåíî áîëüøîå êîëè÷åñòâî ðàáîò (ñì., íàïðèìåð, áèá-ëèîãðàôèþ â [12]).

Ïåðâûì ñòðîãèì è íàèáîëåå ãëóáîêèì èññëåäîâàíèåì òàêèõ óðàâíå-íèé ÿâëÿåòñÿ ïèîíåðñêàÿ ðàáîòà [3] Ñ.Ë. Ñîáîëåâà, ïî èìåíè êîòîðîãîýòè óðàâíåíèÿ è áûëè íàçâàíû.

Äîêëàä ïðîäîëæàåò èññëåäîâàíèÿ, èçëîæåííûå â [4] è êàñàþùèåñÿèçó÷åíèÿ ñâîéñòâ ðåøåíèÿ ïåðâîé íà÷àëüíî-êðàåâîé çàäà÷è äëÿ óðàâ-íåíèé ñîáîëåâñêîãî òèïà ñ ïîñòîÿííûìè è ñ ïåðåìåííûìè êîýôôèöèåí-òàìè.

Ðàññìàòðèâàåìûå çàäà÷è îáúåäèíÿåò ìåòîäèêà èññëåäîâàíèÿ: ñíà-÷àëà âûÿâëÿåòñÿ îãðàíè÷åííîñòü íîðìû ðåøåíèÿ ïîëèíîìîì â ñîîò-âåòñòâóþùåì ïðîñòðàíñòâå Ñîáîëåâà; çàòåì ñ ïîìîùüþ íåêîòîðîãî èí-òåãðàëüíîãî ïðåîáðàçîâàíèÿ ïîäõîäÿùåãî ôóíêöèîíàëüíîãî ïðîñòðàí-ñòâà îïðåäåëÿåòñÿ ïðîñòðàíñòâî ôóíêöèé îäíîé ïåðåìåííîé, îáëàäà-þùèõ íåêîòîðûì êà÷åñòâåííûì ñâîéñòâîì (îñöèëëÿöèÿ; ìîíîòîííûéðîñò; ñòðåìëåíèå ê íóëþ ñ ðîñòîì âðåìåíè).

Èçâåñòíî, ÷òî ðåøåíèå ïåðâîé íà÷àëüíî-êðàåâîé çàäà÷è äëÿ óðàâíå-íèé, íå ÿâëÿþùèõñÿ óðàâíåíèÿìè òèïà Êîøè Êîâàëåâñêîé, íå çàâè-ñèò íåïðåðûâíûì îáðàçîì îò ãðàíèöû ïðîñòðàíñòâåííîé îáëàñòè. Ðàñ-ñìàòðèâàåìûé ìåòîä äàåò âîçìîæíîñòü èññëåäîâàòü ñâîéñòâà ðåøåíèÿ

46


â ïðîñòðàíñòâåííîé îáëàñòè, îïðåäåëÿåìîé ëèøü òðåáîâàíèÿìè ñîîò-âåòñòâóþùèõ òåîðåì âëîæåíèÿ ïðîñòðàíñòâà Ñîáîëåâà â ïðîñòðàíñòâîíåïðåðûâíûõ ôóíêöèé.



2. Ñâåøíèêîâ À. Ã., Àëüøèí À.Á., Êîðïóñîâ Ì.Î., Ïëåòíåð Þ.Ä. Ëèíåéíûåè íåëèíåéíûå óðàâíåíèÿ ñîáîëåâñêîãî òèïà. Ì.: Ôèçìàòëèò, 2007.

3. Ñîáîëåâ Ñ.Ë. Îá îäíîé íîâîé çàäà÷å ìàòåìàòè÷åñêîé ôèçèêè // Èçâ. ÀÍÑÑÑÐ. Ñåð. ìàò. Ò. 18, 1. Ñ. 350.

4. Äåíèñîâà Ò.Å. Àñèìïòîòè÷åñêîå ïîâåäåíèå ðåøåíèÿ ïåðâîé íà÷àëüíî-êðàåâîé çàäà÷è äëÿ óðàâíåíèé ñîáîëåâñêîãî òèïà ñ òî÷êè çðåíèÿ îñöèë-ëÿöèè // Äèôôåðåíö. óðàâíåíèÿ. 2012. Ò. 48, 2. C. 196206.

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Ê ÒÅÎÐÈÈ ÃÐÀÍÈ×ÍÎÉ ÇÀÄÀ×ÈÑÎËÎÍÍÈÊÎÂÀ ÔÀÇÀÍÎ

Äæåíàëèåâ Ì.Ò.1, Èñêàêîâ Ñ.À.2, Ðàìàçàíîâ Ì.È.2

1Èíñòèòóò ìàòåìàòèêè è ìàòåìàòè÷åñêîãîìîäåëèðîâàíèÿ ÌÎÍ ÐÊ, Àëìàòû, Ðåñïóáëèêà Êàçàõñòàí;

[email protected]Èíñòèòóò ïðèêëàäíîé ìàòåìàòèêè, Êàðàãàíäèíñêèé

ãîñóäàðñòâåííûé óíèâåðñèòåò èì. Å.À. Áóêåòîâà, Êàðàãàíäà,Ðåñïóáëèêà Êàçàõñòàí; [email protected], [email protected]

Â äîêëàäå íàðÿäó ñ òðèâèàëüíûì ðåøåíèåì ìû óñòàíàâëèâàåì âêëàññå ñóùåñòâåííî îãðàíè÷åííûõ ôóíêöèé ñ çàäàííûì âåñîì ñóùå-ñòâîâàíèå íåòðèâèàëüíîãî ðåøåíèÿ ñ òî÷íîñòüþ äî ïîñòîÿííîãî ìíîæè-òåëÿ. Êàê îòìå÷åíî â ðàáîòå [1], ñëó÷àé íåîäíîðîäíîé ãðàíè÷íîé çàäà÷è. . . îêàçûâàåòñÿ ïîëåçíûì ïðè èçó÷åíèè íåêîòîðûõ çàäà÷ ñî ñâîáîäíû-ìè ãðàíèöàìè.

Ðàññìàòðèâàåòñÿ îäíîðîäíûé âàðèàíò ãðàíè÷íîé çàäà÷è èç [1]:

ut(x, t)− a2uxx(x, t) = 0, (x, t) ∈ G = (x, t) : 0 < x < kt, t > 0, (1)

ux(x, t)|x=0 = 0, bux(x, t)|x=kt +du(t)

dt= 0, (2)

ãäå u(t) = u(t, t), b = const ≥ 0, k = const > 0.Òåîðåìà. Ãðàíè÷íàÿ çàäà÷à (1)(2) èìååò íàðÿäó ñ òðèâèàëüíûì

ðåøåíèåì è íåòðèâèàëüíîå ðåøåíèå u(x, t) = C u(x, t), u(x, t) ∈L∞,θ(x,t)(G) êëàññ ñóùåñòâåííî îãðàíè÷åííûõ ôóíêöèé ñ âåñîì θ(x, t),

C = const. Çäåñü θ(x, t) = (x + t1/2)−1, åñëè (x, t) ∈ GT , è θ(x, t) =exp

−k(2x+ kt)/(4a2)

, åñëè (x, t) ∈ G \GT , ãäå GT = (x, t) : 0 < x <

kt, 0 < t < T < +∞ ïðîèçâîëüíûé îãðàíè÷åííûé òðåóãîëüíèê.

Ðàáîòà âûïîëíåíà ïðè ïîääåðæêå Êîìèòåòà íàóêè Ìèíèñòåðñòâà îáðàçî-

âàíèÿ è íàóêè Ðåñïóáëèêè Êàçàõñòàí (ïðîåêòû 0085/ÏÖÔ-14, 0823/ÃÔ4

è 1164/ÃÔ4).


1. Ñîëîííèêîâ Â.À., Ôàçàíî À. Îá îäíîìåðíîé ïàðàáîëè÷åñêîé çàäà÷å, âîç-íèêàþùåé ïðè èçó÷åíèè íåêîòîðûõ çàäà÷ ñî ñâîáîäíûìè ãðàíèöàìè //Çàï. íàó÷í. ñåì. ÏÎÌÈ. 2000. Ò. 269. Ñ. 322338.

48


ÑÂÎÉÑÒÂÀ ÃËÜÄÅÐÎÂÎÉ ÍÅÏÐÅÐÛÂÍÎÑÒÈÐÅØÅÍÈÉ ÄÈÔÔÅÐÅÍÖÈÀËÜÍÛÕÍÅÐÀÂÅÍÑÒÂ Ñ ÍÅÊÎÒÎÐÛÌÈÍÓËÜ-ËÀÃÐÀÍÆÈÀÍÀÌÈ

Åãîðîâ À.À.


[email protected], [email protected]

Ðàññìàòðèâàþòñÿ îòîáðàæåíèÿ v : U → Rm êëàññà Ñîáîëåâà W 1p,loc,

p > 1, îïðåäåëåííûå íà îòêðûòîì ìíîæåñòâå U ⊂ Rn è óäîâëåòâîðÿþ-ùèå äèôôåðåíöèàëüíîìó íåðàâåíñòâó

F (v′(x)) ≤ KG(v′(x)) +H(x) äëÿ ï. â. x ∈ U, K > 0, (1)

ãäå G : Rm×n → R k-îäíîðîäíûé íóëü-ëàãðàíæèàí, F : Rm×n → R ôóíêöèÿ, óäîâëåòâîðÿþùàÿ íåðàâåíñòâó F (ζ) ≥ cF |ζ|k äëÿ ζ ∈ Rm×n,cF > 0, H : U → R èçìåðèìàÿ ôóíêöèÿ. Çäåñü k, n, m ∈ N,2 ≤ k ≤ minn,m, v′(x) ìàòðèöà ßêîáè îòîáðàæåíèÿ v â òî÷êå x ∈ U ,Rm×n ïðîñòðàíñòâî âåùåñòâåííûõ m× n-ìàòðèö, ðàññìàòðèâàåìîå ñîïåðàòîðíîé íîðìîé | · |.

Íàéäåíû íîâûå óñëîâèÿ íà ôóíêöèè F , G è H, ñòåïåíü p è êîýôôè-öèåíò K, ãàðàíòèðóþùèå ëîêàëüíóþ ã¼ëüäåðîâîñòü ðåøåíèé íåðàâåí-ñòâà (1). Ïîëó÷åííûå ðåçóëüòàòû óñèëèâàþò ñîîòâåòñòâóþùèå òåîðåìûðàáîò [1, 2].




1. Åãîðîâ À.À. Êâàçèâûïóêëûå ôóíêöèè è íóëü-ëàãðàíæèàíû â ïðîáëåìàõóñòîé÷èâîñòè êëàññîâ îòîáðàæåíèé // Ñèá. ìàò. æóðí. 2008. Ò. 49, 4.Ñ. 796812.

2. Egorov A.A. Solutions of the dierential inequality with a null Lagrangian:higher integrability and removability of singularities. I, II // Âëàäèêàâêàç-ñêèé ìàò. æóðí. 2014. Ò. 16, 3. Ñ. 2237; Ò. 16, 4. Ñ. 4148.

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ÎÖÅÍÈÂÀÍÈÅ ÊÎÝÔÔÈÖÈÅÍÒÎÂÀÂÒÎÍÎÌÍÛÕ ËÈÍÅÉÍÛÕ ÓÐÀÂÍÅÍÈÉ

Åãîðøèí À.Î.


Ðàññìàòðèâàþòñÿ çàäà÷è ñðåäíåêâàäðàòè÷åñêîãî ïðèáëèæåíèÿ ïî-ñëåäîâàòåëüíîñòåé îòñ÷åòîâ íåïðåðûâíûõ ôóíêöèé èç L2(IT ) íà ðàâíî-ìåðíîé h-ñåòêå èëè ôèíèòíûõ ïîñëåäîâàòåëüíîñòåé èç l2(L). Ïðèáëèæå-íèå îñóùåñòâëÿåòñÿ ïåðåõîäíûìè ïðîöåññàìè àâòîíîìíûõ äèôôåðåíöè-àëüíûõ èëè ðàçíîñòíûõ óðàâíåíèé (ÄÓ èëè ÐÓ) íà êîíå÷íîì èíòåðâàëåIT äëèíû T = hL. Ïîêàçàíî: îòñ÷åòû ðåøåíèé ÄÓ ýòîãî êëàññà íà h-ñåòêå ìîãóò áûòü îïèñàíû ðåøåíèÿìè ÐÓ ýòîãî êëàññà. Ýòî ïîçâîëÿåòïî îò÷åòàì ðåøåíèé ÄÓ è çàäàííîé ñåòêå îäíîçíà÷íî îöåíèâàòü êîýô-ôèöèåíòû ÄÓ è ïðèáëèæàòü åãî ðåøåíèÿìè ôóíêöèè èç L2(IT ).

Ïóñòü yk = yik0 ∈ l2(k) âåêòîð èç Ek+1, ãäå ‖y‖2k =∑kj=0 α

∗j |yj |

2.

Ðàññìàòðèâàþòñÿ âàðèàöèîííûå çàäà÷è àïïðîêñèìàöèè: ìèíèìèçèðî-

âàòü Jk = ‖y − y‖2k ïðè óñëîâèè, ÷òî Dk(α)yk =∑n

0 α∗j yj+m

k−nm=0

= 0,

k = 0, L. Ïóñòü S åäèíè÷íàÿ ñôåðà â En+1, à Jk(α) çíà÷åíèå ôóíê-öèîíàëîâ Jk íà ïðîåêöèÿõ yk âåêòîðîâ yk íà ÿäðà îïåðàòîðîâ Dk(α).

Ïîêàçàíî, ÷òî ôóíêöèîíàëû Jk(α) = α∗Qkα èìåþò âèä ïñåâäîêâàäðà-òè÷íûõ ôîðì ñ îáðàçóþùåé ìàòðèöåé Qk = Qk(α). Ïîêàçàíî, ÷òî ïðî-

åêöèè yk, ôóíêöèîíàëû èäåíòèôèêàöèè Jk(α) è èäåíòèôèöèðóþùèåìàòðèöû Qk = Qk(α) ýôôåêòèâíî âû÷èñëÿþòñÿ ñ ïîìîùüþ âñòðå÷íûõóðàâíåíèé äâóñòîðîííåé îðòîãîíàëèçàöèè îäíîðîäíûõ ñèñòåì [1].

Ýêñïåðèìåíòû ïîêàçûâàþò, ÷òî îäíà èòåðàöèÿ íà S âèäà α[1] =

Q−1(α[0])α[0], α[1] = α[1]

/‖α[1]‖, ìîæåò ïðè îïðåäåëåííûõ óñëîâèÿõ äàòü

îöåíêó α[1] òî÷íîãî ðåøåíèÿ α ñ ïðèåìëåìîé äëÿ êîíêðåòíîãî ïðèëî-æåíèÿ ïîãðåøíîñòüþ. Ïîëó÷åíà ñèñòåìà âñòðå÷íûõ óðàâíåíèé, ðåàëè-çóþùàÿ ýòó èòåðàöèþ.




1. Åãîðøèí À.Î. Î âñòðå÷íûõ ïðîöåññàõ îðòîãîíàëèçàöèè // Ñèá. æóðí.âû÷èñë. ìàòåìàòèêè. 2012. Ò. 15, 4. Ñ. 371385.

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Î ÅÄÈÍÑÒÂÅÍÍÎÑÒÈ ÐÅØÅÍÈß ÎÁÐÀÒÍÎÉÇÀÄÀ×È ÑÏÅÊÒÐÀËÜÍÎÃÎ ÀÍÀËÈÇÀ ÄËßÄÈÔÔÅÐÅÍÖÈÀËÜÍÛÕ ÎÏÅÐÀÒÎÐÎÂÂÛÑØÈÕ ÏÎÐßÄÊÎÂ ÍÀ ÎÒÐÅÇÊÅ

Åëåóîâ À.À., Çàêàðèÿíîâà Í.Á., Åëåóîâà Ð.À.

Êàçàõñêèé íàöèîíàëüíûé óíèâåðñèòåò èì. àëü-Ôàðàáè,Àëìàòû, Ðåñïóáëèêà Êàçàõñòàí; [email protected]

Â íàñòîÿùåé ðàáîòå èçëîæåíû íåêîòîðûå ðåçóëüòàòû òåîðèè îáðàò-íûõ ñïåêòðàëüíûõ çàäà÷ äëÿ îáûêíîâåííûõ äèôôåðåíöèàëüíûõ óðàâ-íåíèé. Èçâåñòíî, ÷òî áîëåå òðóäíûìè ÿâëÿþòñÿ îáðàòíûå çàäà÷è äëÿäèôôåðåíöèàëüíûõ óðàâíåíèé âûñøèõ ïîðÿäêîâ ñ íåðàñïàäàþùèìèñÿãðàíè÷íûìè óñëîâèÿìè. Â äàííîé ðàáîòå èññëåäóåòñÿ åäèíñòâåííîñòüðåøåíèÿ îáðàòíîé çàäà÷è ñïåêòðàëüíîãî àíàëèçà äëÿ äèôôåðåíöèàëü-íûõ óðàâíåíèé âûñøèõ ïîðÿäêîâ ñ íåëîêàëüíûìè ãðàíè÷íûìè óñëî-âèÿìè. ×àñòíûé ñëó÷àé óêàçàííûõ ãðàíè÷íûõ óñëîâèé ïðåäñòàâëÿþòäâóõòî÷å÷íûå íåðàñïàäàþùèåñÿ ãðàíè÷íûå óñëîâèÿ. Îñíîâíîé ðåçóëü-òàò íàñòîÿùåé ñòàòüè îáîáùàåò ðåçóëüòàòû ìîíîãðàôèè [1], ãäå ïðèâå-äåíû ïîäîáíûå òåîðåìû åäèíñòâåííîñòè äëÿ ðàñïàäàþùèõñÿ ãðàíè÷íûõóñëîâèé.


1. Íàéìàðê Ì.À. Ëèíåéíûå äèôôåðåíöèàëüíûå îïåðàòîðû. Ì.: Íàóêà, 1969.

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ÇÀÄÀ×À ÎÏÒÈÌÀËÜÍÎÃÎÓÏÐÀÂËÅÍÈß Ñ ÎÁÐÀÒÍÎÉ ÑÂßÇÜÞ

ÄËß ÒÅÐÌÎÂßÇÊÎÓÏÐÓÃÎÉÌÎÄÅËÈ ÊÅËÜÂÈÍÀ ÔÎÉÃÒÀ

Çâÿãèí À.Â.


Ïóñòü Ω ⊂ Rn, n = 2, 3, îãðàíè÷åííàÿ îáëàñòü ñ ãðàíèöåé ∂Ωêëàññà C2. Â QT = [0, T ]× Ω ðàññìàòðèâàåòñÿ çàäà÷à:

∂v

∂t+

n∑i=1

vi∂v

∂xi− 2Div (ν(θ)E)− κ

∂∆v

∂t

−2κDiv

(n∑i=1

vi∂E∂xi

)+∇p ∈ F (v); (1)

div v = 0 â QT ; v|t=0 = v0 â Ω; v|[0,T ]×∂Ω = 0; (2)

∂θ

∂t+

n∑i=1

vi∂θ

∂xi− χ∆ θ

−2

(ν(θ)E + κ

∂E∂t

+ κn∑i=1

vi∂E∂xi

): E(v) ∈ G(θ); (3)

θ|t=0 = θ0 â Ω; θ|[0,T ]×∂Ω = 0. (4)

Çäåñü v, θ è p âåêòîð-ôóíêöèÿ ñêîðîñòè, ôóíêöèè òåìïåðàòóðû èäàâëåíèÿ ñðåäû ñîîòâåòñòâåííî, κ > 0 âðåìÿ ðåòàðäàöèè, χ > 0 êîýôôèöèåíò òåïëîïðîâîäíîñòè, ν(θ) > 0 âÿçêîñòü æèäêîñòè, E òåíçîð ñêîðîñòåé äåôîðìàöèé.

Ðàññìîòðèì ìíîãîçíà÷íûå îòîáðàæåíèÿ

F : E1(:= v : v ∈ L∞([0, T ], V ), v′ ∈ L2(0, T ;V ∗)) ( L2(0, T ;V ∗) è

G : E2(:= v : v ∈ Lp(0, T ;W 1p (Ω)), v′ ∈ L1(0, T ;W−1

p (Ω)), 1 < p < +∞)

( L1(0, T ;W−2(1−1/p)p ),

52


óäîâëåòâîðÿþùèå óñëîâèÿì:(i1) Îòîáðàæåíèÿ F è G îïðåäåëåíû íà ïðîñòðàíñòâàõ E1 è E2 ñîîòâåò-ñòâåííî è èìåþò íåïóñòûå êîìïàêòíûå âûïóêëûå çíà÷åíèÿ;(i2) Îòîáðàæåíèÿ F è G ïîëóíåïðåðûâíû ñâåðõó è êîìïàêòíû;(i3) Îòîáðàæåíèÿ F è G ãëîáàëüíî îãðàíè÷åíû;(i4) Îòîáðàæåíèÿ F è G ñëàáî çàìêíóòû.

Îáîçíà÷èì ÷åðåç Σ ⊂ E1 × E2 ìíîæåñòâî âñåõ ñëàáûõ ðåøåíèé çà-äà÷è (1)(4). Ðàññìîòðèì ôóíêöèîíàë êà÷åñòâà Φ : Σ → R, óäîâëåòâî-ðÿþùèé óñëîâèÿì:(j1) Φ îãðàíè÷åíî ñíèçó;(j2) Åñëè vm v∗ â E1, θm → θ∗ â Lp(0, T ;Lp(Ω)), òî Φ(v∗, θ∗) 6limm→∞

Φ(vm, θm).

Òåîðåìà. Ïóñòü îòîáðàæåíèÿ F è G óäîâëåòâîðÿþò óñëîâèÿì (i1)(i4), à ôóíêöèîíàë Φ (j1)(j2). Òîãäà çàäà÷à îïòèìàëüíîãî óïðàâ-

ëåíèÿ ñ îáðàòíîé ñâÿçüþ (1)(4) èìååò õîòÿ áû îäíî ñëàáîå ðåøåíèå

(v∗, θ∗) òàêîå, ÷òî Φ(v∗, θ∗) = inf(v,θ)∈Σ

Φ(v, θ).

53


ÀÒÒÐÀÊÒÎÐÛ ÌÎÄÅËÈ ÁÈÍÃÀÌÀÑ ÏÅÐÈÎÄÈ×ÅÑÊÈÌÈ ÓÑËÎÂÈßÌÈ

ÏÎ ÏÐÎÑÒÐÀÍÑÒÂÅÍÍÛÌ ÏÅÐÅÌÅÍÍÛÌÂ ÒÐÕÌÅÐÍÎÌ ÑËÓ×ÀÅ

Çâÿãèí Â. Ã.1, Òóðáèí Ì.Â.2

Âîðîíåæñêèé ãîñóäàðñòâåííûé óíèâåðñèòåò, Âîðîíåæ, Ðîññèÿ;1zvg [email protected], [email protected]

Ìîäåëü äâèæåíèÿ ñðåäû Áèíãàìà ïðèìåíÿåòñÿ äëÿ îïèñàíèÿ äâè-æåíèÿ âÿçêîïëàñòè÷íûõ ñðåä. Îíà èññëåäîâàíà ñ òî÷êè çðåíèÿ ñóùå-ñòâîâàíèÿ ðåøåíèé [1]. Àòòðàêòîðû ýòîé ìîäåëè â äâóìåðíîé îáëàñòèèçó÷àëèñü â [2] ñ èñïîëüçîâàíèåì êëàññè÷åñêîé òåîðèè äèíàìè÷åñêèõñèñòåì, äëÿ êîòîðîé íåîáõîäèìûì óñëîâèåì ÿâëÿåòñÿ åäèíñòâåííîñòüðåøåíèÿ ðàññìàòðèâàåìîé çàäà÷è. Îäíàêî äëÿ ñèñòåìû óðàâíåíèé ìî-äåëè Áèíãàìà â òðåõìåðíîì ñëó÷àå åäèíñòâåííîñòü ñëàáûõ ðåøåíèéíåèçâåñòíà. Íàìè äëÿ àâòîíîìíîé ñèòóàöèè äîêàçûâàåòñÿ ñóùåñòâîâà-íèå òðàåêòîðíîãî è ãëîáàëüíîãî àòòðàêòîðîâ â òðåõìåðíîì ñëó÷àå ïðèïåðèîäè÷åñêèõ óñëîâèÿõ ïî ïðîñòðàíñòâåííûì ïåðåìåííûì. Äëÿ äîêà-çàòåëüñòâà ñóùåñòâîâàíèÿ àòòðàêòîðîâ áûëà èñïîëüçîâàíà àáñòðàêòíàÿòåîðèÿ òðàåêòîðíûõ è ãëîáàëüíûõ àòòðàêòîðîâ (ñì. [3]), â êîòîðîé åäèí-ñòâåííîñòè íå òðåáóåòñÿ.

Ðàáîòà âûïîëíåíà ïðè ôèíàíñîâîé ïîääåðæêå Ìèíèñòåðñòâà îáðàçîâàíèÿ

è íàóêè ÐÔ (ïðîåêò 14.Z50.31.0037).


1. Shelukhin V.V. Bingham viscoplastic as a limit of non-Newtonian uids //J. Math. Fluid Mech. 2002. V. 4, No 2. P. 109127.

2. Ñåðåãèí Ã.À. Î äèíàìè÷åñêîé ñèñòåìå, ïîðîæäåííîé äâóìåðíûìè óðàâíå-íèÿìè äâèæåíèÿ ñðåäû Áèíãàìà // Çàï. íàó÷í. ñåì. ËÎÌÈ. 1991. Ò. 188.Ñ. 128142.

3. Zvyagin V.G., Vorotnikov D.A. Topological approximation methods forevolutionary problems of nonlinear hydrodynamics. Berlin, New York: Walterde Gruyter, 2008.

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ÎÁ ÎÄÍÎÉ ÇÀÄÀ×Å ÄËß ÓÐÀÂÍÅÍÈßÒÐÅÒÜÅÃÎ ÏÎÐßÄÊÀ ÑÎÑÒÀÂÍÎÃÎ ÒÈÏÀ

Çèêèðîâ Î.Ñ.

Íàöèîíàëüíûé óíèâåðñèòåò Óçáåêèñòàíà èì. Ì. Óëóãáåêà,Òàøêåíò, Ðåñïóáëèêà Óçáåêèñòàí; [email protected]

Â äàííîé ðàáîòå ðàññìàòðèâàåòñÿ çàäà÷à òèïà Äèðèõëå äëÿ ëèíåé-íîãî óðàâíåíèÿ òðåòüåãî ïîðÿäêà ñîñòàâíîãî òèïà

Mu ≡(α∂

∂x+ β

∂

∂y

)(uxx + uyy) + Lu = g(x, y), (1)

ãäå α, β çàäàííûå ïîñòîÿííûå ÷èñëà, ïðè÷åì α2 + β2 6= 0, à L ëèíåéíîå äèôôåðåíöèàëüíîå âûðàæåíèå âèäà

Lu ≡ a(x, y)uxx+2b(x, y)uxy + c(x, y)uyy +d(x, y)ux+e(x, y)uy +f(x, y)u.

Ïóñòü Ω îäíîñâÿçíàÿ îáëàñòü â ïëîñêîñòè (x, y), îãðàíè÷åííàÿãëàäêèì æîðäàíîâûì êîíòóðîì σ, êîòîðûé îáëàäàåò ñëåäóþùèìè ñâîé-ñòâàìè:

à) âñÿêàÿ ïðÿìàÿ, ïàðàëëåëüíàÿ õàðàêòåðèñòèêå βx − αy = const,ïåðåñåêàåò åãî â äâóõ òî÷êàõ;

á) ïðÿìûå βx − αy = c1 è βx − αy = c2 (c1 < c2) èìåþò ñ íèìåäèíñòâåííûå îáùèå òî÷êè (òî÷êè êàñàíèÿ) M è N ñîîòâåòñòâåííî.

Ðàçîáüåì êðèâóþ σ íà äâå ÷àñòè σ1 è σ2 ñëåäóþùèì îáðàçîì:

σ1 = (x, y) ∈ σ : αxn + βyn > 0, σ2 = σ\σ1,

ãäå xn = cos(n, x), yn = cos(n, y) è n âíåøíÿÿ íîðìàëü ê ãðàíèöå.Äëÿ óðàâíåíèÿ (1) èçó÷àåòñÿ ñëåäóþùàÿ çàäà÷à: òðåáóåòñÿ íàéòè

ôóíêöèþ, óäîâëåòâîðÿþùóþ âíóòðè Ω óðàâíåíèþ (1) è êðàåâûì óñëî-âèÿì:

u(x, y)∣∣σ

= ϕ1(x, y),∂u(x, y)

∂n

∣∣σ2

= ϕ2(x, y),

ãäå ϕ1(x, y) è ϕ2(x, y) çàäàííûå ôóíêöèè.Äîêàçàíû òåîðåìû ñóùåñòâîâàíèÿ è åäèíñòâåííîñòè êëàññè÷åñêî-

ãî ðåøåíèÿ äëÿ ðàññìàòðèâàåìîé çàäà÷è. Äîêàçàòåëüñòâî îñíîâàíî íàýíåðãåòè÷åñêèõ íåðàâåíñòâàõ è íà òåîðèè èíòåãðàëüíûõ óðàâíåíèéôðåäãîëüìîâñêîãî òèïà.

55


ÈÑÑËÅÄÎÂÀÍÈÅ ÍÀ×ÀËÜÍÎ-ÊÐÀÅÂÎÉÇÀÄÀ×È ÄËß ÑÈÑÒÅÌÛ ÓÐÀÂÍÅÍÈÉÍÀÂÜÅ ÑÒÎÊÑÀ Ñ ÏÅÐÅÌÅÍÍÎÉ

ÏËÎÒÍÎÑÒÜÞ

Çîëîòîòðóáîâà Ã.Î.


Ñèñòåìà óðàâíåíèé Íàâüå Ñòîêñà äëÿ æèäêîñòè ñ ïåðåìåííîé ïëîò-íîñòüþ â îãðàíè÷åííîé îáëàñòè Ω ⊂ R3 ñ ëîêàëüíî-ëèïøèöåâîé ãðàíè-öåé Γ èìååò âèä:

ρ

(∂v

∂t+

n∑i=1

vi∂v

∂xi

)− µ∆v + grad p = ρf, (1)

∂ρ

∂t+

n∑i=1

vi∂ρ

∂xi= 0, div v = 0, (2)

v|t=0 = v0, ρ|t=0 = ρ0, v(x, t) = 0 ∀(x, t) ∈ Γ× [0, T ]. (3)

Îïðåäåëåíèå. Ñëàáûì ðåøåíèåì çàäà÷è (1)(3) íàçûâàåòñÿ ïàðà(v, ρ): υ ∈ L2(0, T ;V ) ∩ Cω(0, T ;H), υ′ ∈ L1(0, T ;V ∗), ρ ∈ L∞(0, T ;L∞),0 < m < ρ < M , ãäå m è M êîíñòàíòû, åñëè (v, ρ) óäîâëåòâîðÿåò (3)è äëÿ ëþáûõ ϕ ∈ V , ψ ∈ H1 ïðè ïî÷òè âñåõ t ∈ [0, T ] âûïîëíÿþòñÿ äâàèíòåãðàëüíûõ ðàâåíñòâà:(

ρ(υ)∂υ

∂t, ϕ

)(t) +

n∑i=1

(υiρ

∂υ

∂xi, ϕ

)(t) + µ(∇υ,∇ϕ)(t) = (ρf, ϕ)(t),

(∂ρ

∂t, ψ

)(t) +

n∑i=1

(vi∂ρ

∂xi, ψ

)(t) = 0.

Òåîðåìà. Ïóñòü f ∈ L2(0, T ;V ), υ0 ∈ V , ρ0 ∈ L∞(Ω), 0 < m < ρ <M . Òîãäà ñóùåñòâóåò õîòÿ áû îäíî ñëàáîå ðåøåíèå çàäà÷è (1)(3).

56


ÈÍÒÅÃÐÀËÜÍÎÅ ÏÐÅÄÑÒÀÂËÅÍÈÅ ÐÅØÅÍÈßÎÄÍÎÃÎ ÂÛÐÎÆÄÀÞÙÅÃÎÑß

Â-ÝËËÈÏÒÈ×ÅÑÊÎÃÎ ÓÐÀÂÍÅÍÈßÑ ÏÎËÎÆÈÒÅËÜÍÛÌ ÏÀÐÀÌÅÒÐÎÌ

Èáðàãèìîâà Í.À.

Êàçàíñêèé ãîñóäàðñòâåííûé ýíåðãåòè÷åñêèé óíèâåðñèòåò,Êàçàíü, Ðîññèÿ; [email protected]

Â ðàáîòå ñòðîèòñÿ ôóíäàìåíòàëüíîå ðåøåíèå îäíîãî ìíîãîìåðíîãîâûðîæäàþùåãîñÿ Â-ýëëèïòè÷åñêîãî óðàâíåíèÿ ñ ïîëîæèòåëüíûì ïàðà-ìåòðîì. Äàåòñÿ èíòåãðàëüíîå ïðåäñòàâëåíèå ðåøåíèÿ óðàâíåíèÿ è èçó-÷àþòñÿ ñâîéñòâà ðåøåíèÿ.

Ïóñòü E++p ÷àñòü p-ìåðíîãî åâêëèäîâà ïðîñòðàíñòâà, ãäå xp−1 > 0,

xp > 0, D êîíå÷íàÿ îáëàñòü â E++p , îãðàíè÷åííàÿ ïîâåðõíîñòüþ Γ è

÷àñòÿìè Γ0 è Γ1 ïëîñêîñòåé xp−1 = 0, xp = 0, ñîîòâåòñòâåííî. Îáî-çíà÷èì ÷åðåç x = (x′, xp), x

′ = (x′′, xp−1), x′′ = (x1, x2, . . . , xp−2) òî÷-êè åâêëèäîâà ïðîñòðàíñòâà, à ÷åðåç CkBl(D) ìíîæåñòâî ôóíêöèé, kðàç íåïðåðûâíî äèôôåðåíöèðóåìûõ â D è óäîâëåòâîðÿþùèõ óñëîâèþ∂u∂xl

= o(1) ïðè xl → 0.

Ðàññìîòðèì â E++p âûðîæäàþùååñÿ B-ýëëèïòè÷åñêîå óðàâíåíèå ñ

ïîëîæèòåëüíûì ïàðàìåòðîì âèäà

xmp

(p−2∑l=1

∂2u

∂x2l

+Bxp−1u

)+∂2u

∂x2p

+ λ2xmp u = 0, (1)

ãäå Bxp−1 =∂2

∂x2p−1

+k

xp−1

∂

∂xp−1 îïåðàòîð Áåññåëÿ, m > 0, k > 0,

p > 3, λ ∈ R.Ôóíäàìåíòàëüíûì ðåøåíèåì óðàâíåíèÿ (1) ñ îñîáåííîñòüþ â òî÷êå

x0 ÿâëÿåòñÿ ôóíêöèÿ

Ω(x, x0) = αCk

π∫0

Cγ π∫0

ρ−νϕ H(1)ν (λρϕ) sinγ−1 ϕdϕ

sink−1 ϕdϕ,

ãäå H(1)ν (λρϕ) ôóíêöèÿ Õàíêåëÿ,

ρϕ =

(|x′′ − x′′0 |2 + x2

p−1 + x2p−10

− 2xp−1xp−10 cosϕ

57


+4

(m+ 2)2

(xm+2p + xm+2

p0 − 2xm+2

2p x

m+22

p0 cosϕ))1/2

,

Cγ =Γ( γ+1

2 )√π Γ( γ2 )

, Ck =Γ( k+1

2 )√π Γ( k2 )

, α = 22−ν+γλν

i (m+2)γΓ( γ+12 )Γ( k+1

2 )πp−42

, ν = p+k+γ−22 ,

γ = mm+2 .

Äëÿ ëþáîãî ðåøåíèÿ u(x) èç êëàññà C2Bp−1(D) ∩ C2

Bp(D) ∩ C1(D) èëþáîé òî÷êè x0 ∈ D èìååò ìåñòî ñëåäóþùåå èíòåãðàëüíîå ïðåäñòàâëå-íèå:

u(x0) =

∫Γ

[Ω(ξ, x0)A[u(ξ)]− u(ξ)A[Ω(ξ, x0)]

]ξkp−1dΓ. (2)

Èç èíòåãðàëüíîãî ïðåäñòàâëåíèÿ (2) âûòåêàåò ñëåäóþùåå ñâîéñòâîðåøåíèÿ óðàâíåíèÿ (1):

• ñóùåñòâóåò ðåøåíèå u(x) óðàâíåíèÿ (1) â îáëàñòè De = E++p \D,

óäîâëåòâîðÿþùåå óñëîâèþ

u(x) = O(r−

p+k+γ−12

)ïðè r =

√x2

1 + . . .+ x2p →∞.

58


ÈÍÒÅÃÐÈÐÓÞÙÈÉ ÌÍÎÆÈÒÅËÜÏÀÐÛ ÍÅÇÀÂÈÑÈÌÛÕ ÔÎÐÌ ÏÔÀÔÔÀ

Èâàíîâ Â.Â.


Çäåñü óêàçàíû óñëîâèÿ, íåîáõîäèìûå è äîñòàòî÷íûå äëÿ òîãî, ÷òîáûäâå ãëàäêèå äèôôåðåíöèàëüíûå ôîðìû ω1 è ω2 ïåðâîé ñòåïåíè âáëèçèòî÷êè, ãäå îíè íåçàâèñèìû, èìåëè îáùèé èíòåãðèðóþùèé ìíîæèòåëü.Ýòî òàêàÿ ãëàäêàÿ ïîëîæèòåëüíàÿ ôóíêöèÿ µ, ÷òî ôîðìû µω1 è µω2

ñëóæàò äèôôåðåíöèàëàìè du1 è du2 íåêîòîðûõ ôóíêöèé u1 è u2. Ëåãêîïîíÿòü, ÷òî íàøè ôîðìû äîëæíû óäîâëåòâîðÿòü òðåì óñëîâèÿì

ω1 ∧ dω1 = 0, ω2 ∧ dω2 = 0, ω1 ∧ dω2 + ω2 ∧ dω1 = 0,

èç êîòîðûõ ïåðâûå äâà ýòî êëàññè÷åñêèå óñëîâèÿ Ôðîáåíèóñà ïîëíîéèíòåãðèðóåìîñòè [1] êàæäîãî èç óðàâíåíèé Ïôàôôà ω1 = 0 è ω2 = 0,äëÿ òðåõìåðíîãî ñëó÷àÿ óêàçàííûå åùå Ýéëåðîì. Ïðè ýòèõ è òîëüêîýòèõ òðåõ óñëîâèÿõ ñóùåñòâóåò òàêàÿ ôîðìà Ω, ÷òî ω1 ∧ Ω = dω1 èω2 ∧ Ω = dω2. Òàêàÿ ôîðìà Ω ðîâíî îäíà è ÿâíî âûðàæàåòñÿ ÷åðåçω1 è ω2 ïðè ïîìîùè àðèôìåòè÷åñêèõ äåéñòâèé è äèôôåðåíöèðîâàíèÿ.Ôîðìû ω1 è ω2 èìåþò îáùèé ìíîæèòåëü òîãäà è òîëüêî òîãäà, êîãäàôîðìà Ω òî÷íà, ÷òî äëÿ äâàæäû ãëàäêèõ èñõîäíûõ ôîðì ðàâíîñèëüíîåå çàìêíóòîñòè, è åñëè ýòî òàê, ïðÿìûì èíòåãðèðîâàíèåì ìû íàéäåìôóíêöèþ L, äëÿ êîòîðîé dL = Ω. ßñíî, ÷òî ôóíêöèÿ µ = expL áóäåòîáùèì èíòåãðèðóþùèì ìíîæèòåëåì ôîðì ω1 è ω2, òàê ÷òî ôóíêöèèu1 è u2, ÷üè ìíîæåñòâà óðîâíÿ îïèñûâàþò èíòåãðàëüíûå ìíîãîîáðàçèÿôîðì ω1 è ω2, âûðàæàþòñÿ ÷åðåç ýòè äâå ôîðìû â êâàäðàòóðàõ.


1. Õàðòìàí Ô. Îáûêíîâåííûå äèôôåðåíöèàëüíûå óðàâíåíèÿ. Ì.: Ìèð, 1970.

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ÌÅÒÎÄ ÎÏÒÈÌÀËÜÍÎÃÎ ÓÏÐÀÂËÅÍÈßÂ ÇÀÄÀ×Å ÄËß ÌÀÒÅÌÀÒÈ×ÅÑÊÎÉÌÎÄÅËÈ ÂÇÀÈÌÎÄÅÉÑÒÂÈß ÊËÅÒÎÊÍÎÂÎÎÁÐÀÇÎÂÀÍÈß È ÈÌÌÓÍÈÒÅÒÀ

Â ÓÑËÎÂÈßÕ ÐÀÄÈÎÒÅÐÀÏÈÈ

Êàáàíèõèí Ñ.È.1,2, Êðèâîðîòüêî Î.È.1,2, Êîíäàêîâà Å.À.2

1Èíñòèòóò âû÷èñëèòåëüíîé ìàòåìàòèêè è ìàòåìàòè÷åñêîéãåîôèçèêè ÑÎ ÐÀÍ, Íîâîñèáèðñê, Ðîññèÿ;

2Íîâîñèáèðñêèé ãîñóäàðñòâåííûé óíèâåðñèòåò, Íîâîñèáèðñê, Ðîññèÿ;[email protected], [email protected], [email protected]

×èñëåííî èññëåäîâàíà çàäà÷à Êîøè äëÿ ñèñòåìû íåëèíåéíûõ îáûê-íîâåííûõ äèôôåðåíöèàëüíûõ óðàâíåíèé (ÎÄÓ), êîòîðàÿ õàðàêòåðèçó-åò âçàèìîäåéñòâèå íîâîîáðàçîâàíèé è èììóííûõ êëåòîê â óñëîâèÿõ ðà-äèîòåðàïèè [1]. Ïàðàìåòðû èññëåäóåìîé ìàòåìàòè÷åñêîé ìîäåëè õàðàê-òåðèçóþò ñòåïåíü íîâîîáðàçîâàíèÿ, à òàêæå ñêîðîñòü èììóííîãî îòâåòà.

Â ðàáîòå, îïèðàÿñü íà èññëåäîâàíèÿ ôàçîâûõ ïîðòðåòîâ ñèñòåìû [2],ðàññìàòðèâàþòñÿ âîïðîñû óñòîé÷èâîñòè çàäà÷è Êîøè äëÿ ñèñòåìû íå-ëèíåéíûõ ÎÄÓ. Óñòàíîâëåíî âëèÿíèå ÷ëåíà íà óñòîé÷èâîñòü, îòâå÷àþ-ùåãî çà ëå÷åíèå. Öåëü ðàáîòû ñîñòîèò â ðàçðàáîòêå è èññëåäîâàíèè ÷èñ-ëåííîãî àëãîðèòìà îïðåäåëåíèÿ îïòèìàëüíîãî ëå÷åíèÿ äëÿ ìàòåìàòè-÷åñêîé ìîäåëè âçàèìîäåéñòâèÿ èììóííûõ êëåòîê ñ íîâîîáðàçîâàíèÿìèñ ïåðèîäè÷åñêèì ëå÷åíèåì ðàäèîòåðàïèåé [3, 4]. Ïðèâåäåíû è ïðîàíà-ëèçèðîâàíû ÷èñëåííûå ðàñ÷åòû, äåìîíñòðèðóþùèå ïðåèìóùåñòâî îï-òèìàëüíîãî ëå÷åíèÿ ïåðåä íåýôôåêòèâíûì ïîëíûì ëå÷åíèåì èëè åãîîòñóòñòâèåì.

Ðàáîòà ïðîâîäèëàñü ïðè ÷àñòè÷íîé ïîääåðæêå Ìèíèñòåðñòâà îáðàçîâàíèÿ

è íàóêè Ðîññèéñêîé Ôåäåðàöèè (4.1.3 ñîâìåñòíûå ëàáîðàòîðèè ÍÃÓ ÍÍÖ

ÑÎ ÐÀÍ), ãðàíòîì Ïðåçèäåíòà Ðîññèéñêîé Ôåäåðàöèè ÌÊ-1214.2017.1 è ïðî-

åêòîì ñ Ðåñïóáëèêîé Êàçàõñòàí 1746/ÃÔ4 Òåîðèÿ è ÷èñëåííûå ìåòîäû

ðåøåíèÿ îáðàòíûõ è íåêîððåêòíûõ çàäà÷ åñòåñòâîçíàíèÿ.


1. Liu Z., Yang Ch. A mathematical model of cancer treatment by radio-therapy // Computational and Mathematical Methods in Medicine. 2014.Article ID 172923.

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2. Ãîäóíîâ Ñ.Ê. Ñîâðåìåííûå àñïåêòû ëèíåéíîé àëãåáðû. Íîâîñèáèðñê: Íà-ó÷íàÿ êíèãà, 1997.

3. Êàáàíèõèí Ñ.È. Îáðàòíûå è íåêîððåêòíûå çàäà÷è. Íîâîñèáèðñê: Ñèáèð-ñêîå íàó÷íîå èçäàòåëüñòâî, 2009.

4. Ledzewicz U., Mosalman M. S. F., Schattler H. Optimal controls for a mathe-matical model of tumor-immune interactions under targeted chemotherapywith immune boost // Discrete Contin. Dyn. Syst., Ser. B. 2013. V. 18, No 4.P. 10311051.

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×ÈÑËÅÍÍÎÅ ÐÅØÅÍÈÅÍÀ×ÀËÜÍÎ-ÊÐÀÅÂÎÉ ÇÀÄÀ×È

ÄËß ÓÐÀÂÍÅÍÈß ÝËÅÊÒÐÎÄÈÍÀÌÈÊÈ

Êàáàíèõèí Ñ.È.1, Øèøëåíèí Ì.À.2, Øîëïàíáàåâ Á.Á.3

1Èíñòèòóò âû÷èñëèòåëüíîé ìàòåìàòèêè è ìàòåìàòè÷åñêîéãåîôèçèêè ÑÎ ÐÀÍ, Íîâîñèáèðñê, Ðîññèÿ; [email protected]Èíñòèòóò ìàòåìàòèêè èì. Ñ.Ë. Ñîáîëåâà ÑÎ ÐÀÍ,

Íîâîñèáèðñê, Ðîññèÿ; [email protected]Êàçàõñêèé íàöèîíàëüíûé ïåäàãîãè÷åñêèé óíèâåðñèòåò èì. Àáàÿ,

Àëìàòû, Ðåñïóáëèêà Êàçàõñòàí; [email protected]

Ðàññìîòðèì îáðàòíóþ çàäà÷ó â îáëàñòè Ω = ∆(Lx) × (0, Ly), ãäå∆(Lx) = (x, t) : x ∈ (0, Lx), t ∈ (x, 2Lx − x):

utt +σ√µ

√εut = uxx + uyy, (x, t) ∈ ∆(Lx), (1)

ux(0, y, t) = g(y, t), y ∈ (0, Ly), t ∈ (0, 2Lx), (2)

u(x, y, x) = q(x, y), x ∈ (0, Lx), y ∈ (0, Ly), (3)

u(x, 0, t) = u(x, Ly, t) = 0, (x, t) ∈ ∆(Lx). (4)

Â ïðÿìîé çàäà÷å (1)(4) òðåáóåòñÿ îïðåäåëèòü u(x, y, t) ïî çàäàí-íûì q(x, y) è g(y, t).

Îáðàòíàÿ çàäà÷à çàêëþ÷àåòñÿ â îïðåäåëåíèè ôóíêöèè q(x, y) èçñîîòíîøåíèé (1)(4) ïî äîïîëíèòåëüíîé èíôîðìàöèè:

u(0, y, t) = f(y, t).

Àëãîðèòì ðåøåíèÿ îáðàòíîé çàäà÷è

1. Âûáèðàåì íà÷àëüíîå ïðèáëèæåíèå q0.

2. Ðåøàåì ïðÿìóþ çàäà÷ó (1)(4) ñ çàäàííûì qn.

3. Âû÷èñëÿåì çíà÷åíèå ôóíêöèîíàëà

J(qn) =Ly∫0

2Lx∫0

[u(0, y, t; qn)− f(y, t)]2dydt.

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4. Åñëè çíà÷åíèå öåëåâîãî ôóíêöèîíàëà íå äîñòàòî÷íî ìàëî, òîãäàðåøàåì ñîïðÿæåííóþ çàäà÷ó.

5. Âû÷èñëÿåì ãðàäèåíò ôóíêöèîíàëà

J ′qn = ψt(x, y, x) +σ√µ

√εψ(x, y, x).

6. Âû÷èñëÿåì ñëåäóþùåå ïðèáëèæåíèå qn+1 = qn − αJ ′qn è ïåðåõî-äèì ê ïóíêòó 2.

Ðàáîòà âûïîëíåíà ïðè ôèíàíñîâîé ïîääåðæêå Êîìèòåòà íàóêè ÌÎÍ ÐÊ

1746/ÃÔ4 Òåîðèÿ è ÷èñëåííûå ìåòîäû ðåøåíèÿ îáðàòíûõ è íåêîððåêòíûõ

çàäà÷ åñòåñòâîçíàíèÿ.


1. Êàáàíèõèí Ñ.È., Íóðñåèòîâ Ä.Á., Øîëïàíáàåâ Á.Á. Çàäà÷à ïðîäîëæå-íèÿ ýëåêòðîìàãíèòíîãî ïîëÿ â ñòîðîíó çàëåãàíèÿ íåîäíîðîäíîñòåé // Ñèá.ýëåêòðîí. ìàò. èçâ. 2014. Ò. 11. Òðóäû V Ìåæäóíàðîäíîé ìîëîäåæíîéøêîëû-êîíôåðåíöèè Òåîðèÿ è ÷èñëåííûå ìåòîäû ðåøåíèÿ îáðàòíûõ èíåêîððåêòíûõ çàäà÷. Ñ. C85C102.

2. Kabanikhin S. I., Nurseitov D.B., Shishlenin M.A., Sholpanbaev B.B. Inverseproblems for the ground penetrating radar // J. Inverse Ill-Posed Probl. 2013.V. 21, No 6. P. 885892.

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ÊÐÀÅÂÛÅ ÇÀÄÀ×È ÄËß ÍÅÊÎÒÎÐÛÕÊËÀÑÑÎÂ ÑÈËÜÍÎ ÍÅËÈÍÅÉÍÛÕÓÐÀÂÍÅÍÈÉ ÑÎÁÎËÅÂÑÊÎÃÎ ÒÈÏÀ

Êîæàíîâ À.È.


Â äîêëàäå èçó÷àþòñÿ ðåçóëüòàòû î ðàçðåøèìîñòè â öåëîì â êëàñ-ñàõ ðåãóëÿðíûõ (èìåþùèõ âñå îáîáùåííûå ïî Ñ.Ë. Ñîáîëåâó ïðîèç-âîäíûå, âõîäÿùèå â óðàâíåíèå) ðåøåíèé íà÷àëüíî-êðàåâûõ çàäà÷ äëÿñèëüíî íåëèíåéíûõ óðàâíåíèé ñîáîëåâñêîãî òèïà. Áîëåå êîíêðåòíî: áó-äóò ïðåäñòàâëåíû ðåçóëüòàòû î ðàçðåøèìîñòè íà÷àëüíî-êðàåâûõ çàäà÷äëÿ:

à) ñèëüíî íåëèíåéíûõ óðàâíåíèé ñîáîëåâñêîãî òèïà íå÷åòíîãî ïî-ðÿäêà (ïñåâäîãèïåðáîëè÷åñêèõ è ïñåâäîïàðàáîëè÷åñêèõ óðàâíåíèé),âîçíèêàþùèõ â òåîðèè âÿçêîóïðóãèõ ñðåä;

á) íåëèíåéíûõ àíàëîãîâ óðàâíåíèÿ Áóññèíåñêà Ëÿâà, âîçíèêàþùèõâ ýëåêòðîäèíàìèêå;

â) íåëèíåéíûõ èíòåãðîäèôôåðåíöèàëüíûõ óðàâíåíèé, ñâÿçàííûõ ñóðàâíåíèÿìè ñîáîëåâñêîãî òèïà.

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ÎÄÍÎÌÅÐÍÎÅ ÂÎËÍÎÂÎÅ ÓÐÀÂÍÅÍÈÅÄËß ÍÅÎÄÍÎÐÎÄÍÎÉ ÑÐÅÄÛ

Êîíîâàëîâà Ä.Ñ.


Ïðè îòñóòñòâèè âíåøíåé ñèëû óðàâíåíèå ïîïåðå÷íûõ êîëåáàíèéñòðóíû è ïðîäîëüíûõ êîëåáàíèé ñòåðæíÿ çàïèñûâàåòñÿ â âèäå:

∂2u

∂t2− a2 ∂

2u

∂x2= 0. (1)

Ê óðàâíåíèþ (1) ïðèñîåäèíÿþòñÿ äàííûå Êîøè:

u(x, 0) = ϕ(x),∂u(x, 0)

∂t= ψ(x). (2)

Ôóíêöèÿ ϕ(x) èìååò íåïðåðûâíóþ âòîðóþ ïðîèçâîäíóþ, à ôóíêöèÿψ(x) ïåðâóþ. Ôóíêöèÿ a(x) ïðåäïîëàãàåòñÿ êóñî÷íî-ïîñòîÿííîé:a(x) = a1 ïðè x ≤ x0, a(x) = a2 ïðè x > x0, a1, a2 ïîëîæèòåëüíûå÷èñëà. Â ïîëóïëîñêîñòè t > 0 îïðåäåëÿþòñÿ îáëàñòè Gi, i = 1, . . . , 4, ïî-ëó÷åííûå ðàçäåëåíèåì ïîëóïðîñòðàíñòâà t > 0 òðåìÿ ëó÷àìè, èñõîäÿ-ùèìè èç òî÷êè (x0, 0). Ðåøåíèå óðàâíåíèÿ (1) ïîíèìàåòñÿ â ñëåäóþùåìîáîáùåííîì ñìûñëå: îíî íåïðåðûâíî, óäîâëåòâîðÿåò ñîîòíîøåíèÿì (2)è â êàæäîé îáëàñòè Gi èìååò íåïðåðûâíûå ïðîèçâîäíûå utt, uxx, óäî-âëåòâîðÿþùèå óðàâíåíèþ (1). Íà âûøåóêàçàííûõ ëó÷àõ äîïóñêàþòñÿðàçðûâû ïåðâîãî ðîäà ïðîèçâîäíûõ îáîáùåííîãî ðåøåíèÿ.

Èññëåäîâàíèå çàäà÷è ïðîèçâîäèòñÿ ñâåäåíèåì óðàâíåíèÿ (1) ê ñè-ñòåìå äâóõ óðàâíåíèé ïåðâîãî ïîðÿäêà ñ ðàçðûâíûìè êîýôôèöèåíòàìèïðè ïðîèçâîäíîé ïî x. Ñèñòåìà ðåøàåòñÿ ìåòîäîì õàðàêòåðèñòèê, è âèòîãå ïîëó÷àåòñÿ ôîðìóëà äëÿ u(x, t), êîòîðóþ ìîæíî íàçâàòü îáîáùåí-íîé ôîðìóëîé Äàëàìáåðà. Èçó÷åííàÿ çàäà÷à ÿâëÿåòñÿ íåáîëüøèì ôðàã-ìåíòîì òåîðèè çîíäèðîâàíèÿ íåîäíîðîäíûõ ñðåä ôèçè÷åñêèìè ñèãíàëà-ìè.

Ðàáîòà âûïîëíåíà ïðè ïîääåðæêå ïðîãðàììû Ïðåçèäèóìà ÐÀÍ (ïðîåêò

0314-2015-0010).

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Î ÐÎÁÀÑÒÍÎÉ ÓÑÒÎÉ×ÈÂÎÑÒÈÑÒÀÖÈÎÍÀÐÍÛÕ ÄÈÔÔÅÐÅÍÖÈÀËÜÍÎ-

ÀËÃÅÁÐÀÈ×ÅÑÊÈÕ ÓÐÀÂÍÅÍÈÉÑÎ ÑÒÐÓÊÒÓÐÈÐÎÂÀÍÍÎÉÍÅÎÏÐÅÄÅËÅÍÍÎÑÒÜÞ

Êîíîíîâ À.Ä.

Èíñòèòóò äèíàìèêè ñèñòåì è òåîðèè óïðàâëåíèÿèì. Â.Ì. Ìàòðîñîâà ÑÎ ÐÀÍ, Èðêóòñê, Ðîññèÿ;

[email protected]

Ðàññìàòðèâàåòñÿ ñòàöèîíàðíàÿ ñèñòåìà äèôôåðåíöèàëüíûõ óðàâíå-íèé

Ax′(t) + (B + C∆D)x(t) = 0, t ∈ T = [0,+∞), (1)

ñî ñòðóêòóðèðîâàííîé íåîïðåäåëåííîñòüþ C∆D. Çäåñü A, B, C è D n × n çàäàííûå ìàòðèöû, ïó÷îê λA+B ðåãóëÿðåí, detA = 0. Òàêèåñèñòåìû íàçûâàþòñÿ äèôôåðåíöèàëüíî-àëãåáðàè÷åñêèìè óðàâíåíèÿìè(ÄÀÓ). Ìåðîé íåðàçðåøåííîñòè ÄÀÓ îòíîñèòåëüíî ïðîèçâîäíîé ñëó-æèò öåëî÷èñëåííàÿ âåëè÷èíà, íàçûâàåìàÿ èíäåêñîì.

Èññëåäóåòñÿ âîïðîñ îá àñèìïòîòè÷åñêîé óñòîé÷èâîñòè ñèñòåìû (1) âóñëîâèÿõ, êîãäà àñèìïòîòè÷åñêè óñòîé÷èâà íåâîçìóùåííàÿ ñèñòåìà

Ax′(t) +Bx(t) = 0, t ∈ T. (2)

Àíàëèç ïðîâîäèòñÿ â ïðåäïîëîæåíèÿõ, îáåñïå÷èâàþùèõ ñóùåñòâî-âàíèå îïåðàòîðà R = R0 +R1

ddt + · · ·+Rr

(ddt

)r, äåéñòâèå êîòîðîãî ïðå-

îáðàçóåò ñèñòåìó (2) ê âèäó(0 0En−d 0

)(x′1(t)x′2(t)

)+

(J1 EdJ2 0

)(x1(t)x2(t)

)= 0, t ∈ T,

ãäå r èíäåêñ íåðàçðåøåííîñòè, Ed åäèíè÷íàÿ ìàòðèöà óêàçàííîãî

ïîðÿäêà;

(x1(t)x2(t)

)= Qx(t), Q ìàòðèöà ïåðåñòàíîâêè ñòðîê; J1 è

J2 íåêîòîðûå ìàòðèöû ñîîòâåòñòâóþùèõ ðàçìåðîâ [1].Îñíîâíàÿ ñëîæíîñòü, âîçíèêàþùàÿ ïðè èññëåäîâàíèè ðîáàñòíûõ

ñâîéñòâ ÄÀÓ, ñâÿçàíà ñ òåì, ÷òî â ñëó÷àå âûñîêîãî èíäåêñà ïðè âîç-ìóùåíèè âõîäíûõ äàííûõ ìîæåò èçìåíèòüñÿ âíóòðåííÿÿ ñòðóêòóðàñèñòåìû.

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Äëÿ ñèñòåìû (1) èíäåêñîâ íåðàçðåøåííîñòè 1 è 2 â óñëîâèÿõ, ïðèêîòîðûõ ñòðóêòóðèðîâàííîå âîçìóùåíèå C∆D íå ìåíÿåò ñòðóêòóðó ñè-ñòåìû, ïîëó÷åíû äîñòàòî÷íûå óñëîâèÿ ðîáàñòíîé óñòîé÷èâîñòè.

Ðàáîòà âûïîëíåíà ïðè ÷àñòè÷íîé ôèíàíñîâîé ïîääåðæêå Ðîññèéñêîãî

ôîíäà ôóíäàìåíòàëüíûõ èññëåäîâàíèé (ïðîåêò 16-31-00101) è Êîìïëåêñ-

íîé ïðîãðàììû ôóíäàìåíòàëüíûõ íàó÷íûõ èññëåäîâàíèé ÑÎ ÐÀÍ II.2.


1. Ùåãëîâà À.À. Ñóùåñòâîâàíèå ðåøåíèÿ íà÷àëüíîé çàäà÷è äëÿ âûðîæäåí-íîé ëèíåéíîé ãèáðèäíîé ñèñòåìû ñ ïåðåìåííûìè êîýôôèöèåíòàìè // Èçâ.âóçîâ. Ìàòåì. 2010. 9. Ñ. 5770.

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Ê ÂÎÏÐÎÑÓ ÎÁ ÀÍÀËÈÒÈ×ÅÑÊÎÉÐÀÇÐÅØÈÌÎÑÒÈ ÇÀÄÀ× Ñ ÂÛÐÎÆÄÅÍÈÅÌ

ÄËß ÍÅËÈÍÅÉÍÎÃÎ ÓÐÀÂÍÅÍÈßÒÅÏËÎÏÐÎÂÎÄÍÎÑÒÈ

Êóçíåöîâ Ï.À.

Èíñòèòóò ìàòåìàòèêè, ýêîíîìèêè è èíôîðìàòèêè ÈÃÓ,Èðêóòñê, Ðîññèÿ; [email protected]

Ïðåäñòàâëåíû ðåçóëüòàòû èññëåäîâàíèÿ ðÿäà ñïåöèàëüíûõ êðàåâûõçàäà÷ âèäà

ut = u∆u+1

σ(∇u)2, u|Γ = α(t, x) (1)

äëÿ íåëèíåéíîãî óðàâíåíèÿ òåïëîïðîâîäíîñòè/ôèëüòðàöèè (the porousmedium equation) â êëàññå àíàëèòè÷åñêèõ ôóíêöèé. Çäåñü u = u(t, x) íåèçâåñòíàÿ ôóíêöèÿ, x ∈ Rn, n = 2, 3, σ > 0 êîíñòàíòà. Çàìêíóòàÿäîñòàòî÷íî ãëàäêàÿ ïîâåðõíîñòü Γ îãðàíè÷èâàåò çâåçäíóþ îáëàñòü âRn. Àíàëèòè÷åñêàÿ ôóíêöèÿ α îáðàùàåòñÿ â íóëü ïðè t = 0, ÷òî äåëàåòóðàâíåíèå (1) íå ðàçðåøåííûì îòíîñèòåëüíî ñòàðøåé ïðîèçâîäíîé [1].

Äëÿ çàäà÷è (1) ïîñëåäîâàòåëüíî ðàññìîòðåíû ñëåäóþùèå âèäû ñòà-öèîíàðíîé ãðàíèöû Γ: òðåõìåðíàÿ ñôåðà [2], ïðîèçâîëüíàÿ äâóìåðíàÿêðèâàÿ [3], ïðîèçâîëüíàÿ òðåõìåðíàÿ ïîâåðõíîñòü. Â êàæäîì ñëó÷àåïîñòðîåíî ðåøåíèå â âèäå äâîéíîãî ñòåïåííîãî ðÿäà, à òàêæå äîêàçàíàñîîòâåòñòâóþùàÿ òåîðåìà ñóùåñòâîâàíèÿ. Ðåçóëüòàò èç [3] îáîáùåí íàñëó÷àé íåñòàöèîíàðíîé Γ ≡ Γ(t).

Ðàáîòà âûïîëíåíà ïðè ÷àñòè÷íîé ïîääåðæêå Ðîññèéñêîãî ôîíäà ôóíäà-

ìåíòàëüíûõ èññëåäîâàíèé (ïðîåêòû 16-01-00608 è 16-31-00291).



2. Êàçàêîâ À.Ë., Êóçíåöîâ Ï.À. Êðàåâàÿ çàäà÷à ñ âûðîæäåíèåì äëÿ íåëè-íåéíîãî óðàâíåíèÿ òåïëîïðîâîäíîñòè â ñôåðè÷åñêèõ êîîðäèíàòàõ // Òåçè-ñû äîêëàäîâ Ìåæäóíàðîäíîé êîíôåðåíöèè Äèôôåðåíöèàëüíûå óðàâíå-íèÿ. Ôóíêöèîíàëüíûå ïðîñòðàíñòâà. Òåîðèÿ ïðèáëèæåíèé. Íîâîñèáèðñê:Èçä-âî ÈÌ ÑÎ ÐÀÍ, 2013. C. 149.

3. Êàçàêîâ À.Ë., Êóçíåöîâ Ï.À. Îá îäíîé êðàåâîé çàäà÷å äëÿ íåëèíåéíîãîóðàâíåíèÿ òåïëîïðîâîäíîñòè â ñëó÷àå äâóõ ïðîñòðàíñòâåííûõ ïåðåìåí-íûõ // Ñèá. æóðí. èíäóñòð. ìàòåìàòèêè. 2014. Ò. 17, 1. Ñ. 4654.

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ÀÑÈÌÏÒÎÒÈÊÀ ÐÅØÅÍÈßÍÅÑÒÀÖÈÎÍÀÐÍÎÉ ÑÈÍÃÓËßÐÍÎ-ÂÎÇÌÓÙÅÍÍÎÉ ÊÐÀÅÂÎÉ ÇÀÄÀ×È

ÒÅÏËÎÌÀÑÑÎÎÁÌÅÍÀ

Ëàòûïîâ È.È.

Áèðñêèé ôèëèàë Áàøêèðñêîãî ãîñóäàðñòâåííîãî óíèâåðñèòåòà,Áèðñê, Ðîññèÿ; [email protected]

Â äîêëàäå ñòàâèòñÿ è ðåøàåòñÿ çàäà÷à íàõîæäåíèÿ ïðèáëèæåííîãîðåøåíèÿ ñèíãóëÿðíî âîçìóùåííîé êðàåâîé çàäà÷è óðàâíåíèÿ òåïëîïðî-âîäíîñòè ñ íåëèíåéíûìè ãðàíè÷íûìè óñëîâèÿìè íà ïîäâèæíûõ ãðàíè-öàõ [1, 2]:

Tt (x, t) = Fo · Txx (x, t) + q (x, t) +A [T (x, t)] , T (x, t) = T0 (x) , t 0,

Tx (x, t) = −qi (t) + γi ·[T 4 (x, t)− U4

i

], x = ψi (t) , i = 1, 2,

(x, t) ∈ Ω = (x, t) : ψ1 (t) < x < ψ2 (t) , 0 < t ≤ t0 , 0 < Fo 1,

q1 (t) = q10 · exp

− (t− β0)

2

β102

, q2 (t) = q20 = const.

Ïðèáëèæåííîå ðåøåíèå, èñïîëüçóÿ ãåîìåòðî-îïòè÷åñêèé àñèìïòî-òè÷åñêèé ìåòîä [2], ïîëó÷àåòñÿ â âèäå àñèìïòîòè÷åñêîãî ðàçëîæåíèÿðåøåíèÿ â ñìûñëå Ïóàíêàðå ïî ñòåïåíÿì ìàëûõ ïàðàìåòðîâ â çàâèñè-ìîñòè îò áëèçîñòè ðàññìàòðèâàåìîé òî÷êè ê ãðàíèöàì [3].


1. Ëàòûïîâ È.È. Ïðèáëèæåííûé ðàñ÷åò ðàñïðåäåëåíèÿ òåìïåðàòóðíîãî ïî-ëÿ àêòèâíîãî ýëåìåíòà òâåðäîòåëüíîãî ëàçåðà // Òðóäû êàôåäðû ýêñïå-ðèìåíòàëüíîé è òåîðåòè÷åñêîé ôèçèêè Èíñòèòóòà ôèçèêè ìîëåêóë è êðè-ñòàëëîâ ÓÍÖ ÐÀÍ. Âûï. 1. Óôà: Ãèëåì, 2001. Ñ. 8292.

2. Ëàòûïîâ È.È., Êðàâ÷åíêî Â.Ô., Íåñåíåíêî Ã.À. Ïðèìåíåíèå èíòåãðàëü-íûõ óðàâíåíèé ê ñèíãóëÿðíî âîçìóùåííîé íåñòàöèîíàðíîé êðàåâîé çàäà÷åòåïëîïðîâîäíîñòè ñ ïîäâèæíûìè ãðàíèöàìè // Äèôôåðåíö. óðàâíåíèÿ.1999. Ò. 35, 9. Ñ. 11711178.

3. Ëàòûïîâ È.È., Áèãàåâà Ë.À. Èññëåäîâàíèå íåñòàöèîíàðíûõ òåïëîâûõïðîöåññîâ â ìàòåðèàëå ïðè âîçäåéñòâèè óëüòðàêîðîòêèìè ëàçåðíûìè èì-ïóëüñàìè // Êîíöåïöèè ôóíäàìåíòàëüíûõ è ïðèêëàäíûõ íàó÷íûõ èññëå-äîâàíèé. Ñáîðíèê ñòàòåé Ìåæäóíàðîäíîé íàó÷íî-ïðàêòè÷åñêîé êîíôå-ðåíöèè. ×. 3. Óôà: Îìåãà Ñàéíñ, 2016. Ñ. 1115.

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ÌÅÒÎÄÛ ÏÐÀÊÒÈ×ÅÑÊÎÉÈÄÅÍÒÈÔÈÖÈÐÓÅÌÎÑÒÈ ÍÅÊÎÒÎÐÛÕ

ÌÀÒÅÌÀÒÈ×ÅÑÊÈÕ ÌÎÄÅËÅÉ ÁÈÎËÎÃÈÈ

Ëàòûøåíêî Â.À.1, Êðèâîðîòüêî Î.È.2, Êàáàíèõèí Ñ.È.2

1Íîâîñèáèðñêèé ãîñóäàðñòâåííûé óíèâåðñèòåò, Íîâîñèáèðñê, Ðîññèÿ;[email protected]

2Èíñòèòóò âû÷èñëèòåëüíîé ìàòåìàòèêè è ìàòåìàòè÷åñêîéãåîôèçèêè ÑÎ ÐÀÍ, Íîâîñèáèðñê, Ðîññèÿ;[email protected], [email protected]

Â äàííîé ðàáîòå èññëåäîâàíû íåêîòîðûå ìåòîäû ïðàêòè÷åñêîé (àïî-ñòåðèîðíîé) èäåíòèôèöèðóåìîñòè ïàðàìåòðîâ ìàòåìàòè÷åñêèõ ìîäåëåéáèîëîãèè [1]. Äàííûå ìåòîäû ïîçâîëÿþò îïðåäåëèòü åäèíñòâåííûé íà-áîð ïàðàìåòðîâ ìàòåìàòè÷åñêèõ ìîäåëåé áèîëîãèè ïî ðåçóëüòàòàì èç-ìåðåíèÿ îïðåäåëåííûõ âûõîäíûõ âåëè÷èí â òå÷åíèå îïòèìàëüíîãî èí-òåðâàëà âðåìåíè.

Öåëü äàííîé ðàáîòû âûÿâèòü íåèäåíòèôèöèðóåìûå ïàðàìåòðûíåêîòîðûõ ìàòåìàòè÷åñêèõ ìîäåëåé áèîëîãèè è íàéòè îïòèìàëüíîå êî-ëè÷åñòâî èçìåðåíèé, êîòîðîå íåîáõîäèìî áðàòü â òå÷åíèå íåêîòîðîãîèíòåðâàëà âðåìåíè, ÷òîáû äîñòàòî÷íî òî÷íî îïðåäåëÿòü ïàðàìåòðû ìà-òåìàòè÷åñêîé ìîäåëè. Äëÿ ýòîãî èñïîëüçóåòñÿ ìåòîä ñîáñòâåííûõ çíà-÷åíèé è ìåòîä, îñíîâàííûé íà àíàëèçå ÷èñëà îáóñëîâëåííîñòè ìàòðèöû÷óâñòâèòåëüíîñòè. Ïðèâåäåíû è ïðîàíàëèçèðîâàíû ðåçóëüòàòû ÷èñëåí-íûõ ðàñ÷åòîâ.

Ðàáîòà âûïîëíåíà ïðè ôèíàíñîâîé ïîääåðæêå Ìèíèñòåðñòâà îáðàçîâàíèÿ

è íàóêè Ðîññèéñêîé Ôåäåðàöèè, ãðàíòà Ïðåçèäåíòà ÐÔ ( ÌÊ-1214.2017.1)

è Ðåñïóáëèêè Êàçàõñòàí (ãðàíò 1746/ÃÔ4 Òåîðèÿ è ÷èñëåííûå ìåòîäû

ðåøåíèÿ îáðàòíûõ è íåêîððåêòíûõ çàäà÷ åñòåñòâîçíàíèÿ).


1. Miao H., Xia X., Perelson A. S., Wu H. On identiability of nonlinear ODEmodels and applications in viral dynamics // SIAM Rev. 2011. V. 53, No 1.P. 339.

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ÎÁ ÓÑÒÎÉ×ÈÂÎÑÒÈ ÍÀ ÊÎÍÅ×ÍÎÌÈÍÒÅÐÂÀËÅ ÂÐÅÌÅÍÈ ÓÒÎÊ-ÖÈÊËÎÂÎÄÍÎÉ ÊÈÍÅÒÈ×ÅÑÊÎÉ ÌÎÄÅËÈ

Ëàøèíà Å.À.1,3, ×óìàêîâà Í.À.1,3, ×óìàêîâ Ã.À.2,3

1Èíñòèòóò êàòàëèçà èì. Ã.Ê. Áîðåñêîâà ÑÎ ÐÀÍ,Íîâîñèáèðñê, Ðîññèÿ;

2Èíñòèòóò ìàòåìàòèêè èì. Ñ.Ë. Ñîáîëåâà ÑÎ ÐÀÍ,Íîâîñèáèðñê, Ðîññèÿ;

3Íîâîñèáèðñêèé ãîñóäàðñòâåííûé óíèâåðñèòåò, Íîâîñèáèðñê, Ðîññèÿ;[email protected], [email protected], [email protected]

Ðàáîòà ïîñâÿùåíà àíàëèçó ïåðèîäè÷åñêèõ ðåøåíèé îäíîé êèíåòè÷åñ-êîé ìîäåëè ãåòåðîãåííîé êàòàëèòè÷åñêîé ðåàêöèè, êîòîðàÿ ÿâëÿåòñÿñèñòåìîé äâóõ íåëèíåéíûõ îáûêíîâåííûõ äèôôåðåíöèàëüíûõ óðàâíå-íèé [1]. Èññëåäîâàíèå ìîäåëè ïðîâîäèòñÿ â ñëó÷àå, êîãäà â íåé âîçìîæ-íî âûäåëåíèå ìàëîãî ïàðàìåòðà 0 < ε 1, è ñèñòåìà ÿâëÿåòñÿ ñèí-ãóëÿðíî âîçìóùåííîé. Ïðè íåêîòîðûõ çíà÷åíèÿõ ïàðàìåòðîâ (ñì. [1])ñîñóùåñòâóþò óñòîé÷èâûé è íåóñòîé÷èâûé ïðåäåëüíûå öèêëû, êîòîðûåîáëàäàþò âûñîêîé ïàðàìåòðè÷åñêîé ÷óâñòâèòåëüíîñòüþ è ÿâëÿþòñÿ óò-êàìè (ñì. [2]). ×èñëåííîå èññëåäîâàíèå óòêè-öèêëà äîñòàòî÷íî ñëîæ-íàÿ çàäà÷à, ò. ê. ìàëûå âîçìóùåíèÿ íà÷àëüíîé òî÷êè ìîãóò âûçûâàòüêîíå÷íûå âàðèàöèè ðåøåíèÿ ïðè óñëîâèè, ÷òî âðåìÿ íàõîæäåíèÿ ôàçî-âîé òî÷êè âáëèçè íåóñòîé÷èâîé ÷àñòè ìåäëåííîãî ìíîãîîáðàçèÿ äîñòà-òî÷íî ïðîäîëæèòåëüíî. Äëÿ àíàëèçà òî÷íîñòè ÷èñëåííîãî èíòåãðèðîâà-íèÿ â ðàáîòå ïðîâîäèòñÿ èññëåäîâàíèå óñòîé÷èâîñòè ðåøåíèé ñèñòåìûíà êîíå÷íîì èíòåðâàëå âðåìåíè è ïðåäëàãàþòñÿ îöåíêè ãëîáàëüíîé ïî-ãðåøíîñòè ÷èñëåííîãî èíòåãðèðîâàíèÿ.

Ðàáîòà âûïîëíåíà â ðàìêàõ ãîñóäàðñòâåííîãî çàäàíèÿ ÔÃÁÓÍ ÈÊ ÑÎ

ÐÀÍ (ïðîåêò 0303-2016-0003).


1. Chumakov G.A., Lashina E.A.,Chumakova N.A. On estimation of the globalerror of numerical solution on canard-cycles // Math. Comput. Simul. 2015.V. 116. P. 5974.

2. Çâîíêèí À.Ê., Øóáèí Ì.À. Íåñòàíäàðòíûé àíàëèç è ñèíãóëÿðíûå âîçìó-ùåíèÿ îáûêíîâåííûõ äèôôåðåíöèàëüíûõ óðàâíåíèé // Óñïåõè ìàò. íàóê.1984. Ò. 39, âûï. 2. Ñ. 77127.

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ÓÑËÎÂÈß ÑÒÀÁÈËÈÇÀÖÈÈÐÅØÅÍÈÉ ÊÐÀÅÂÛÕ ÇÀÄÀ×ÄËß ÓÐÀÂÍÅÍÈß ÑÎÁÎËÅÂÀ

Ëîìàêèí À.Â.


Â ðàáîòå ðàññìàòðèâàþòñÿ çàäà÷à Êîøè è ñìåøàííûå êðàåâûå çà-äà÷è â ÷åòâåðòè ïðîñòðàíñòâà äëÿ óðàâíåíèÿ Ñîáîëåâà [1] ïðè n = 2

∆utt + ux2x2= 0, t > 0.

Èññëåäîâàíû àñèìïòîòè÷åñêèå ñâîéñòâà ðåøåíèé ïðè t → ∞ äëÿ ïðî-èçâîëüíûõ íà÷àëüíûõ äàííûõ. Äîêàçàíî, ÷òî ðåøåíèÿ çàäà÷ â ïðåäåëåâûõîäÿò íà ôóíêöèè, çàâèñÿùèå òîëüêî îò x1. Äëÿ êàæäîé èç çàäà÷ïðåäåëüíûå ôóíêöèè óêàçàíû â ÿâíîì âèäå, óñòàíîâëåíû ñêîðîñòè ñòà-áèëèçàöèè. Ïîëó÷åíû óñëîâèÿ íà íà÷àëüíûå äàííûå, ïðè êîòîðûõ ðå-øåíèÿ ñòðåìÿòñÿ ê íóëþ.

Ïðè ïîëó÷åíèè ðåçóëüòàòîâ ïðèìåíÿëñÿ âàðèàíò ìåòîäà ñòàöèîíàð-íîé ôàçû, ïðåäëîæåííûé â [2, ãë. 3].

Àâòîð âûðàæàåò áëàãîäàðíîñòü ïðîô. Ã.Â. Äåìèäåíêî çà ïîñòàíîâêóçàäà÷è è íàó÷íîå ðóêîâîäñòâî.


1. Ñîáîëåâ Ñ.Ë. Îá îäíîé íîâîé çàäà÷å ìàòåìàòè÷åñêîé ôèçèêè // Èçâ. ÀÍÑÑÑÐ. Ñåð. ìàò. 1954. Ò. 18, 1. Ñ. 350.

2. Óñïåíñêèé Ñ.Â., Äåìèäåíêî Ã.Â., Ïåðåïåëêèí Â. Ã. Òåîðåìû âëîæåíèÿè ïðèëîæåíèÿ ê äèôôåðåíöèàëüíûì óðàâíåíèÿì. Íîâîñèáèðñê: Íàóêà,1984.

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ÎÖÅÍÊÈ ÐÅØÅÍÈÉ ÂÒÎÐÎÉ ÊÐÀÅÂÎÉÇÀÄÀ×È ÄËß ÓÐÀÂÍÅÍÈß ÑÎÁÎËÅÂÀ

Ëîìàêèíà Å.À.


Â ðàáîòå ðàññìàòðèâàåòñÿ âòîðàÿ êðàåâàÿ çàäà÷à äëÿ óðàâíåíèÿ Ñî-áîëåâà [1]

∆utt + ux2x2= 0 (1)

â öèëèíäðè÷åñêîé îáëàñòè Q = t > 0, x ∈ G ⊂ R2. Ïðåäïîëàãàåò-ñÿ, ÷òî îáëàñòü G îãðàíè÷åííàÿ è çâåçäíàÿ îòíîñèòåëüíî íåêîòîðîãîêðóãà.

Îäíîé èç ïðîáëåì Ñ.Ë. Ñîáîëåâà, ïîñòàâëåííîé èì äëÿ óðàâíå-íèÿ (1), ÿâëÿåòñÿ èçó÷åíèå àñèìïòîòè÷åñêèõ ñâîéñòâ ðåøåíèé êðàåâûõçàäà÷ ïðè t→∞. Áîëüøîé âêëàä â ðåøåíèå ýòîé ïðîáëåìû âíåñëè åãîó÷åíèêè Ð.À. Àëåêñàíäðÿí, Ð.Ò. Äåí÷åâ, Ò.È. Çåëåíÿê, Â.Í. Ìàñëåí-íèêîâà è äð. ïðè èçó÷åíèè êðàåâûõ çàäà÷ â îáëàñòÿõ ñïåöèàëüíîãî âèäà.Â ñëó÷àå îáëàñòåé ïðîèçâîëüíîé êîíôèãóðàöèè îöåíêè íà áåñêîíå÷íî-ñòè ðåøåíèÿ ïåðâîé êðàåâîé çàäà÷è äëÿ óðàâíåíèÿ Ñîáîëåâà áûëè ïî-ëó÷åíû â êîíöå ïðîøëîãî ñòîëåòèÿ â [2] ïðè n = 2 è â [3] ïðè n ≥ 2. Èçýòèõ ðåçóëüòàòîâ, â ÷àñòíîñòè, âûòåêàåò, ÷òî ïðè t→∞ ðåøåíèå è åãîïðîèçâîäíûå íå ìîãóò ðàñòè áûñòðåå íåêîòîðîé ñòåïåíè tγ .

Â íàñòîÿùåé ðàáîòå, ñëåäóÿ ñõåìå [3], ìû ïîëó÷àåì àíàëîãè÷íûåîöåíêè ðåøåíèÿ âòîðîé êðàåâîé çàäà÷è.

Àâòîð âûðàæàåò áëàãîäàðíîñòü ïðîô. Ã.Â. Äåìèäåíêî çà ïîñòàíîâêóçàäà÷è è íàó÷íîå ðóêîâîäñòâî.


1. Ñîáîëåâ Ñ.Ë. Îá îäíîé íîâîé çàäà÷å ìàòåìàòè÷åñêîé ôèçèêè // Èçâ. ÀÍÑÑÑÐ. Ñåð. ìàò. 1954. Ò. 18, 1. Ñ. 350.

2. Ñêàçêà Â.Â. Àñèìïòîòèêà ïðè t→∞ ñìåøàííûõ çàäà÷ äëÿ îäíîãî óðàâ-íåíèÿ ìàòåìàòè÷åñêîé ôèçèêè // Ñèá. ìàò. æóðí. 1981. Ò. 22, 1. Ñ. 129143.

3. Äåìèäåíêî Ã.Â. Îöåíêè ïðè t → ∞ ðåøåíèÿ îäíîé çàäà÷è Ñ.Ë. Ñîáîëå-âà // Ñèá. ìàò. æóðí. 1984. Ò. 25, 2. Ñ. 112120.

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ÈÑÑËÅÄÎÂÀÍÈÅ ÑÂÎÉÑÒÂ ÎÖÅÍÎÊÏÀÐÀÌÅÒÐÎÂ ÐÀÇÍÎÑÒÍÛÕ ÓÐÀÂÍÅÍÈÉ ÍÀÊÎÍÅ×ÍÛÕ ÂÛÁÎÐÊÀÕ ËÈÍÅÀÐÈÇÀÖÈÅÉ

ÃÐÀÄÈÅÍÒÀ ÖÅËÅÂÛÕ ÔÓÍÊÖÈÉ

Ëîìîâ À.À.


[email protected]

Ðàññìàòðèâàåòñÿ ñèñòåìà ðàçíîñòíûõ óðàâíåíèé

x[k + 1] = Aθx[k] +Bθu[k] + w[k], k = 1, N − 1,

z[k].=

[x[k]u[k]

]=

[x[k]u[k]

]+

[ξx[k]ξu[k]

].= z[k] + ξ[k],

ãäå x[k] ∈ Rn, u[k] ∈ Rm ïåðåìåííûå, Aθ ∈ Rn×n, Bθ ∈ Rn×m ìàòðèöû, çàâèñÿùèå îò ôèêñèðîâàííîãî ïàðàìåòðà θ ∈ Rv, ïîäëåæàùå-ãî èäåíòèôèêàöèè ïî íàáëþäåíèÿì z[k] ∈ Rn+m. Âåêòîðû w[k] è ξ[k]èãðàþò ðîëü íåçàâèñèìûõ ñëó÷àéíûõ âîçìóùåíèé.

Äëÿ öåëåâûõ ôóíêöèé J1(θ) =(W1θ + W2

)∗Y Y ∗

(W1θ + W2

)[1],

J2(θ) = ‖z − zopt(θ)‖2K# , zopt(θ).= arg minz:Gθz=0 ‖z − z‖2K# [2], ãäå

W1,2.= ΦFW1,2, Φ, F ìàòðèöû äèñêðåòíîãî ïðåîáðàçîâàíèÿ Ôóðüå

è ïðåôèëüòðàöèè, W1,2 ìàòðèöû èç íàáëþäåíèé, Y.= ΦW1(z) [1],

èññëåäóåòñÿ ÷óâñòâèòåëüíîñòü îöåíîê θ1,2 = arg minθ J1,2(θ) ê ìàëûìâîçìóùåíèÿì w[k] è ξ[k].




1. Klein V., Morelli E.A. Aircraft system identication. Theory and practice //AIAA Education Series. Reston: AIAA, 2006.

2. Ëîìîâ À.À. Î ëîêàëüíîé óñòîé÷èâîñòè â çàäà÷å èäåíòèôèêàöèè êîýôôè-öèåíòîâ ëèíåéíîãî ðàçíîñòíîãî óðàâíåíèÿ // Âåñòí. ÍÃÓ. Ñåð. Ìàòåìà-òèêà, ìåõàíèêà, èíôîðìàòèêà. 2010. Ò. 10, âûï. 4. C. 81103.

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ÎÖÅÍÊÈ ×ÓÂÑÒÂÈÒÅËÜÍÎÑÒÈÈÄÅÍÒÈÔÈÊÀÖÈÈ ÊÎÝÔÔÈÖÈÅÍÒÎÂÐÀÇÍÎÑÒÍÛÕ ÓÐÀÂÍÅÍÈÉ ×ÅÐÅÇ

ËÎÊÀËÜÍÛÅ ÐÀÇËÎÆÅÍÈßÖÅËÅÂÛÕ ÔÓÍÊÖÈÉ

Ëîìîâ À.À.1, Ôåäîñååâ À.Â.2



Â äîêëàäå èññëåäóþòñÿ ñâîéñòâà òðåõ ìåòîäîâ èäåíòèôèêàöèè ïà-ðàìåòðîâ ðàçíîñòíûõ óðàâíåíèé â óñëîâèÿõ íåòèïè÷íûõ âîçìóùåíèé âíàáëþäåíèÿõ ñ èñïîëüçîâàíèåì òåîðèè ÷óâñòâèòåëüíîñòè ïî ëîêàëüíûìðàçëîæåíèÿì öåëåâûõ ôóíêöèé. Òåîðåòè÷åñêèå ðåçóëüòàòû ïðîâåðÿþò-ñÿ âû÷èñëèòåëüíûì ìîäåëèðîâàíèåì íà ïðèìåðå óðàâíåíèé ïðîäîëü-íîãî äâèæåíèÿ ëåòàòåëüíîãî àïïàðàòà, ïàðàìåòðû êîòîðîãî èäåíòèôè-öèðóþòñÿ ëèíåéíûì ìåòîäîì íàèìåíüøèõ êâàäðàòîâ, ìåòîäîì èíñòðó-ìåíòàëüíûõ ïåðåìåííûõ â ÷àñòîòíîé îáëàñòè, âàðèàöèîííûì ìåòîäîì(STLS, GTLS).

Èññëåäîâàíèå ñâîéñòâ îöåíîê ïàðàìåòðîâ ðàçíîñòíûõ óðàâíåíèé ïó-òåì ëèíåàðèçàöèè ãðàäèåíòà öåëåâîé ôóíêöèè èìååò ðÿä îãðàíè÷åíèé,êàê ïðèíöèïèàëüíûõ (íåâîçìîæíîñòü èçó÷èòü ñìåùåíèå îöåíîê), òàê èñâÿçàííûõ ñ íåäîñòàòî÷íûì ðàçâèòèåì òåîðèè, â ÷àñòíîñòè, ñ ÷ðåçâû-÷àéíî ñèëüíûìè ïðåäïîëîæåíèÿìè î ìàëîñòè âîçìóùåíèé. Òåì íå ìå-íåå, â äîêëàäå ïîêàçàíî, ÷òî ëèíåàðèçîâàííûå ìîäåëè çàâèñèìîñòè îò-êëîíåíèé îöåíîê îò âîçìóùåíèé äåìîíñòðèðóþò õîðîøåå ñîâïàäåíèå ñðåçóëüòàòàìè âû÷èñëèòåëüíîãî ýêñïåðèìåíòà íà ðåàëüíûõ äàííûõ. Ýòîäàåò îñíîâàíèÿ èñïîëüçîâàòü ëîêàëüíûé ïîäõîä â êà÷åñòâå èíñòðóìåíòàèññëåäîâàíèÿ ñâîéñòâ îöåíîê íà êîíå÷íûõ âûáîðêàõ.



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ÍÅËÎÊÀËÜÍÛÅ ÊÐÀÅÂÛÅ ÇÀÄÀ×ÈÑ ×ÀÑÒÈ×ÍÎ ÈÍÒÅÃÐÀËÜÍÛÌÈ

ÓÑËÎÂÈßÌÈ ÄËß ÂÛÐÎÆÄÀÞÙÈÕÑßÄÈÔÔÅÐÅÍÖÈÀËÜÍÛÕ ÓÐÀÂÍÅÍÈÉ

Ëóêèíà Ã.À.

Ïîëèòåõíè÷åñêèé èíñòèòóò (ôèëèàë) Ñåâåðî-Âîñòî÷íîãîôåäåðàëüíîãî óíèâåðñèòåòà èì. Ì.Ê. Àììîñîâà, Ìèðíûé, Ðîññèÿ;

[email protected]

Èññëåäóåòñÿ ðàçðåøèìîñòü íåëîêàëüíûõ çàäà÷ ñ èíòåãðàëüíûìè ãðà-íè÷íûìè óñëîâèÿìè ïî ïðîñòðàíñòâåííîé ïåðåìåííîé äëÿ âûðîæäàþ-ùèõñÿ óðàâíåíèé c êðàòíûìè õàðàêòåðèñòèêàìè.

Ïóñòü Ω åñòü èíòåðâàë (0, 1) îñè Ox, Q åñòü ïðÿìîóãîëüíèê Ω×(0, T ),0<T <+∞, N(x), K(t), α(t) çàäàííûå ôóíêöèè, îïðåäåëåííûå ïðèx ∈ Ω, t ∈ [0, T ].

Íåëîêàëüíàÿ çàäà÷à I. Íàéòè ôóíêöèþ u(x, t), ÿâëÿþùóþñÿ âïðÿìîóãîëüíèêå Q ðåøåíèåì óðàâíåíèÿ

K(t)ut + uxxx + c(x, t)u = f(x, t) (1)

è òàêóþ, ÷òî äëÿ íåå âûïîëíÿþòñÿ óñëîâèÿ:

u(x, 0) = 0, x ∈ Ω, (2)

u(0, t) = u(1, t) = 0, 0 < t < T, (3)

ux(1, t) =

1∫0

N(x)u(x, t)dx, 0 < t < T.

Íåëîêàëüíàÿ çàäà÷à II. Íàéòè ôóíêöèþ u(x, t), ÿâëÿþùóþñÿ âïðÿìîóãîëüíèêå Q ðåøåíèåì óðàâíåíèÿ (1) è òàêóþ, ÷òî äëÿ íåå âû-ïîëíÿþòñÿ óñëîâèÿ (2), (3), à òàêæå óñëîâèå

ux(1, t) = α(t)ux(0, t), 0 < t < T.

Äëÿ ðàññìàòðèâàåìûõ çàäà÷ äîêàçûâàþòñÿ òåîðåìû ñóùåñòâîâàíèÿðåãóëÿðíûõ ðåøåíèé.

76


ÏÅÐÈÎÄÈ×ÅÑÊÎÅ ÐÅØÅÍÈÅÈÍÒÅÃÐÀËÜÍÎÃÎ ÓÐÀÂÍÅÍÈß

ÀÁÅËß ÏÅÐÂÎÃÎ ÐÎÄÀ

Ìàëþòèíà Ì.Â.1, Îðëîâ Ñ.Ñ.2

Èðêóòñêèé ãîñóäàðñòâåííûé óíèâåðñèòåò, Èðêóòñê, Ðîññèÿ;[email protected], [email protected]

Ðàññìîòðèì èíòåãðàëüíîå óðàâíåíèå

1

Γ(α)

x∫0

(x− t)α−1ϕ(t)dt = f(x), x ≥ 0, (1)

îòíîñèòåëüíî íåèçâåñòíîé ôóíêöèè ϕ, â êîòîðîì Γ(α) ãàììà-ôóíêöèÿÝéëåðà ôèêñèðîâàííîãî àðãóìåíòà α > 0, f çàäàííàÿ ôóíêöèÿ. Ýòîóðàâíåíèå ïðè 0 < α < 1 èìååò ñëàáóþ (èíòåãðèðóåìóþ) îñîáåííîñòü, âñîâðåìåííîé ìàòåìàòè÷åñêîé ëèòåðàòóðå îíî íàçûâàåòñÿ îáîáùåííûìèíòåãðàëüíûì óðàâíåíèåì Àáåëÿ. Êëàññû óðàâíåíèé Àáåëÿ ñîõðàíè-ëè àêòóàëüíîñòü êàê îáúåêòû èññëåäîâàíèÿ, èì ïîñâÿùåíî ìíîæåñòâîæóðíàëüíûõ ïóáëèêàöèé è äàæå îòäåëüíûå ìîíîãðàôèè, íàïðèìåð [1].Ïðåæäå âñåãî, âûñîêèé èíòåðåñ âûçâàí çàïðîñàìè åñòåñòâåííûõ íàóê,ãäå óðàâíåíèÿ Àáåëÿ íàõîäÿò øèðîêîå ïðèìåíåíèå [1, ñ. 2660]. Â òîæå âðåìÿ îí ïðîäèêòîâàí èíòåíñèâíûì ðàçâèòèåì íàïðàâëåíèé ñàìîéìàòåìàòèêè, â äàííîì ñëó÷àå äðîáíîãî èíòåãðî-äèôôåðåíöèðîâàíèÿ,èìåíóåìîãî çà ðóáåæîì êàê fractional calculus.

Íàñòîÿùèé äîêëàä ïîñâÿùåí ïðîáëåìå ñóùåñòâîâàíèÿ íåïðåðûâíûõïåðèîäè÷åñêèõ ðåøåíèé óðàâíåíèÿ (1), êîòîðîé óäåëåíî íåçàñëóæåííîìàëîå âíèìàíèå â ñîâðåìåííîé íàó÷íîé ëèòåðàòóðå, ïðè÷åì êàñàåòñÿýòî êàê óðàâíåíèé Àáåëÿ, òàê è èíòåãðàëüíûõ óðàâíåíèé Âîëüòåððà âöåëîì. Äîêàçàíû êðèòåðèè ñóùåñòâîâàíèÿ åäèíñòâåííîãî íåïðåðûâíîãîíà ëó÷å ïåðèîäè÷åñêîãî ðåøåíèÿ óðàâíåíèÿ (1) äëÿ α ∈ N è α /∈ N.




1. Goreno R., Vessella S. Abel integral equations: analysis and applications.Berlin, Heidelberg: Springer-Verlag, 1991.

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ÃËÎÁÀËÜÍÀß ÐÀÇÐÅØÈÌÎÑÒÜÍÀ×ÀËÜÍÎ-ÊÐÀÅÂÎÉ ÇÀÄÀ×È ÄËß

ÎÄÍÎÌÅÐÍÛÕ ÓÐÀÂÍÅÍÈÉ ÄÈÍÀÌÈÊÈÌÍÎÃÎÑÊÎÐÎÑÒÍÛÕ ÑÌÅÑÅÉ

Ìàìîíòîâ À. Å.1, Ïðîêóäèí Ä.À.2

Èíñòèòóò ãèäðîäèíàìèêè èì. Ì.À. Ëàâðåíòüåâà ÑÎ ÐÀÍ,Íîâîñèáèðñê, Ðîññèÿ;


Ðàññìàòðèâàåòñÿ ìîäåëü ìíîãîêîìïîíåíòíûõ ìíîãîñêîðîñòíûõ ñìå-ñåé. Äëÿ íåå òåîðåìû î ãëîáàëüíîì ñóùåñòâîâàíèè ñëàáûõ ðåøåíèé ìíî-ãîìåðíûõ çàäà÷ áûëè ïîëó÷åíû ñîâñåì íåäàâíî, áëàãîäàðÿ ÷åìó ñîñòîÿ-íèå ýòîé òåîðèè ñòàëî ñîïîñòàâèìûì ñ òàêîâûì äëÿ îäíîêîìïîíåíòíûõìîäåëåé. Ïðè ýòîì îòêðûëèñü ïðîáëåìû, õàðàêòåðíûå èìåííî äëÿ ñìå-ñåé, è îòëè÷àþùèå èõ ïðèíöèïèàëüíî îò îäíîêîìïîíåíòíûõ ìîäåëåé.Â îïðåäåëåííîé ñòåïåíè ýòè òðóäíîñòè îïèñàíû â [1, 2]. Äëÿ ìíîãèõ èçíèõ ïîêà íåÿñíû ïóòè ïðåîäîëåíèÿ.

Êàê è â ìíîãîìåðíîì ñëó÷àå, êëàññè÷åñêèå ðåçóëüòàòû äëÿ îäíîêîì-ïîíåíòíîãî âÿçêîãî ãàçà íå ïåðåíîñÿòñÿ íà ìíîãîêîìïîíåíòíûé îäíî-ìåðíûé ñëó÷àé êàêèì-òî àâòîìàòè÷åñêèì îáðàçîì, â ÷àñòíîñòè, â ñèëóïðèíöèïèàëüíî èíîé ñòðóêòóðû âÿçêèõ ÷ëåíîâ: íàëè÷èÿ íåäèàãîíàëü-íîé ìàòðèöû âÿçêîñòåé. Ýòî îòëè÷èå ïî ñâîåé ñëîæíîñòè íå çàâèñèò îòðàçìåðíîñòè äâèæåíèÿ.

Â äîêëàäå áóäåò ïðåäñòàâëåíà òåîðåìà î ñóùåñòâîâàíèè è åäèíñòâåí-íîñòè ñèëüíîãî ðåøåíèÿ íà÷àëüíî-êðàåâîé çàäà÷è äëÿ îäíîìåðíûõóðàâíåíèé äâèæåíèÿ ìíîãîêîìïîíåíòíûõ ìíîãîñêîðîñòíûõ ñìåñåé ñíåäèàãîíàëüíîé ìàòðèöåé âÿçêîñòåé, áóäóò îáñóæäàòüñÿ ïåðñïåêòèâûè òðóäíîñòè äàëüíåéøåãî ðàçâèòèÿ òåîðèè.

Ðàáîòà âûïîëíåíà ïðè ïîääåðæêå Ðîññèéñêîãî íàó÷íîãî ôîíäà (ïðîåêò

15-11-20019).


1. Mamontov A.E., Prokudin D.A. Viscous compressible multi-uids: modelingand multi-D existence // Methods Appl. Anal. 2013. V. 20, No 2. P. 179195.

2. Mamontov A.E., Prokudin D.A. Viscous compressible homogeneous multi-uids with multiple velocities: barotropic existence theory // Sib. Electron.Math. Reports. 2017. V. 14. P. 388397.

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ÐÎÁÀÑÒÍÀß ÓÑÒÎÉ×ÈÂÎÑÒÜ ÐÅØÅÍÈÉËÈÍÅÉÍÛÕ ÏÅÐÈÎÄÈ×ÅÑÊÈÕ ÑÈÑÒÅÌ

ÍÅÉÒÐÀËÜÍÎÃÎ ÒÈÏÀÑ ÍÅÑÊÎËÜÊÈÌÈ ÇÀÏÀÇÄÛÂÀÍÈßÌÈ

Ìàòâååâà È.È.


[email protected]

Â ðàáîòå èññëåäóåòñÿ ðîáàñòíàÿ óñòîé÷èâîñòü ðåøåíèé ñèñòåì äèô-ôåðåíöèàëüíûõ óðàâíåíèé íåéòðàëüíîãî òèïà ñëåäóþùåãî âèäà

d

dty(t) = A(t)y(t) +

m∑j=1

Bj(t)y(t− τj) +

m∑j=1

Cj(t)d

dty(t− τj), t ≥ 0,

ãäå A(t), Bj(t), Cj(t) ìàòðèöû ðàçìåðà n×n ñ íåïðåðûâíûìè ïåðèîäè-÷åñêèìè ýëåìåíòàìè, τj > 0 ïàðàìåòðû çàïàçäûâàíèÿ, j = 1, . . . ,m.

Ïðåäïîëàãàÿ ýêñïîíåíöèàëüíóþ óñòîé÷èâîñòü íóëåâîãî ðåøåíèÿ íå-âîçìóùåííîé ñèñòåìû, óêàçûâàþòñÿ óñëîâèÿ íà âîçìóùåíèÿ ìàòðèöA(t), Bj(t), Cj(t), ïðè êîòîðûõ íóëåâîå ðåøåíèå âîçìóùåííîé ñèñòåìûýêñïîíåíöèàëüíî óñòîé÷èâî. Äëÿ ðåøåíèé âîçìóùåííîé ñèñòåìû óñòà-íîâëåíû îöåíêè, õàðàêòåðèçóþùèå ñêîðîñòü óáûâàíèÿ íà áåñêîíå÷íî-ñòè. Ïðè äîêàçàòåëüñòâå èñïîëüçóåòñÿ ôóíêöèîíàë Ëÿïóíîâà Êðàñîâ-ñêîãî ñïåöèàëüíîãî âèäà [1].




1. Ìàòâååâà È.È. Îá ýêñïîíåíöèàëüíîé óñòîé÷èâîñòè ðåøåíèé ïåðèîäè÷å-ñêèõ ñèñòåì íåéòðàëüíîãî òèïà ñ íåñêîëüêèìè çàïàçäûâàíèÿìè // Äèô-ôåðåíö. óðàâíåíèÿ. 2017. Ò. 53, 6. Ñ. 730740.

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ÍÅÊÎÐÐÅÊÒÍÛÅ ÇÀÄÀ×È ÌÅÕÀÍÈÊÈ

Ìèðåíêîâ Â. Å.

Èíñòèòóò ãîðíîãî äåëà èì. Í.À. ×èíàêàëà ÑÎ ÐÀÍ,Íîâîñèáèðñê, Ðîññèÿ; [email protected]

Ðàññìàòðèâàþòñÿ íåêîððåêòíûå çàäà÷è ìåõàíèêè äåôîðìèðóåìîãîòâåðäîãî òåëà äëÿ êóñî÷íî-îäíîðîäíûõ îáëàñòåé ñî çíà÷èòåëüíûì ÷èñ-ëîì óïðàâëÿþùèõ ïàðàìåòðîâ, êîòîðûå èäåíòèôèöèðóþòñÿ, èñïîëüçóÿíàòóðíûå äàííûå î ñìåùåíèÿõ ãðàíèöû.

Ãðàíè÷íûå çàäà÷è ìåõàíèêè ÿâëÿþòñÿ ìíîãîïàðàìåòðè÷åñêèìè èçàâèñèìîñòü èñêîìûõ ðåøåíèé îò íèõ ìîæåò áûòü ñëîæíîé. Õàðàêòåðçàâèñèìîñòè ðåøåíèÿ îò ïàðàìåòðîâ ñòàíîâèòñÿ èçâåñòíûì ïîñëå îáðà-ùåíèÿ ãðàíè÷íîé çàäà÷è. Èññëåäîâàíèå ïîâåäåíèÿ ðåøåíèÿ, êàê ïðàâè-ëî, äîñòèãàåòñÿ ÷èñëåííûìè ìåòîäàìè, ïðè ýòîì âñòà¼ò ðÿä âîïðîñîâ îò-íîñèòåëüíî òî÷íîñòè ðåøåíèÿ, îïèñàíèÿ âëèÿíèÿ ïàðàìåòðîâ íà ðåøå-íèå ïðè îäíîâðåìåííîì èõ èçìåíåíèè. Â íàñòîÿùåå âðåìÿ èçâåñòíû ÷èñ-ëåííûå ðåøåíèÿ ÷àñòíûõ îáðàòíûõ çàäà÷ â çàâèñèìîñòè îò èäåíòèôèêà-öèè îäíîãî ïàðàìåòðà. Ïðåäëàãàåòñÿ ïðèíöèïèàëüíî íîâûé ìåòîä ðåøå-íèÿ îáðàòíûõ ìíîãîïàðàìåòðè÷åñêèõ çàäà÷, îñíîâàííûé íà ïîëó÷åííûõòî÷íûõ óðàâíåíèÿõ, ñâÿçûâàþùèõ ãðàíè÷íûå çíà÷åíèÿ êîìïîíåíò íà-ïðÿæåíèé è ñìåùåíèé è èñêëþ÷àþùèõ ðåãóëÿðèçàöèþ. Èñïîëüçîâàíèåýêñïåðèìåíòàëüíûõ äàííûõ, îïðåäåëåííûõ ñ ïîãðåøíîñòüþ, äèñêðåòè-çàöèÿ ñïëîøíîé ñðåäû ïðè ÷èñëåííîì ñ÷¼òå, àïðèîðíûå ïðåäïîëîæå-íèÿ íà õàðàêòåð äåôîðìèðîâàíèÿ êîíñòðóêöèè (èäåàëüíîå ïðîñêàëüçû-âàíèå, ñêà÷îê ñìåùåíèé, íàðóøåíèå êîíôîðìíîñòè â êîíå÷íîì ÷èñëåòî÷åê è ò. ï.) âíîñÿò ïîãðåøíîñòü â ãðàíè÷íûå óñëîâèÿ ïðè ôîðìóëè-ðîâêå çàäà÷è è ðàñøèðÿþò êëàññ îáðàòíûõ çàäà÷. Ïðåäëîæåíà ñèñòå-ìà ñèíãóëÿðíûõ èíòåãðàëüíûõ óðàâíåíèé, ïîçâîëÿþùàÿ ïîëó÷èòü àíà-ëèòè÷åñêèå ðåøåíèÿ äëÿ ïðîèçâîëüíîé îáëàñòè êîìïîíåíò ñìåùåíèéè íàïðÿæåíèé â êâàäðàòóðàõ, ÷òî äà¼ò âîçìîæíîñòü èñêëþ÷èòü ïðî-öåññ ðåãóëÿðèçàöèè è îïðåäåëèòü íà ñòàäèè ôîðìóëèðîâêè ëþáîé èççàäà÷: êîððåêòíà, åñëè ñèñòåìà ñâîäèòñÿ ê óðàâíåíèÿì òèïà Ôðåäãîëü-ìà âòîðîãî ðîäà; íåêîððåêòíà, åñëè òèïà Ôðåäãîëüìà ïåðâîãî ðîäà.Äëÿ ïðîâåäåíèÿ íàòóðíûõ èçìåðåíèé ïðîöåññà äåôîðìèðîâàíèÿ äîñòà-òî÷íî çàìåðÿòü ñìåùåíèÿ íà äîñòóïíûõ ó÷àñòêàõ ãðàíèöû èññëåäóå-ìîé îáëàñòè (äîïîëíèòåëüíàÿ èíôîðìàöèÿ, íåîáõîäèìàÿ äëÿ îáðàòíûõçàäà÷). Àëãîðèòì ðåøåíèÿ ïðåäñòàâëÿåò ñîáîé ïåðåáîðíûé ïðîöåññ, ñ

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ïîìîùüþ êîòîðîãî âîññòàíàâëèâàþòñÿ ïàðàìåòðû, íàèëó÷øèì îáðàçîìóäîâëåòâîðÿþùèå çàäàííûì óñëîâèÿì. Ïðåäëàãàåìûé ìåòîä îñíîâàí íàðåøåíèè îäíîé îòäåëüíîé îáðàòíîé çàäà÷è äëÿ êàæäîãî èç âàðüèðóå-ìûõ ïàðàìåòðîâ. Óñëîâèå ïîëó÷åíèÿ ðåøåíèÿ ñ äîñòàòî÷íîé òî÷íîñòüþïîñëåäîâàòåëüíûìè ïðèáëèæåíèÿìè ïðèâîäèò ê âûïîëíåíèþ áîëüøîãîîáú¼ìà âû÷èñëåíèé. Ïîêàçàíî, ÷òî íååäèíñòâåííîñòü ðåøåíèÿ ìíîãî-ïàðàìåòðè÷åñêèõ íåêîððåêòíûõ çàäà÷ íà ïåðâûõ øàãàõ ïðèáëèæåíèéìîæåò áûòü ñâÿçàíà ñ ïîñëåäîâàòåëüíîñòüþ âûáðàííîãî ïóòè ïðèáëè-æåíèÿ. Âñÿ ïîñëåäîâàòåëüíîñòü â êîíå÷íîì ñ÷¼òå ñõîäèòñÿ ê åäèíñòâåí-íîìó ðåøåíèþ, åñëè íå ñòàâèòü âîïðîñ î êîëè÷åñòâå èòåðàöèé, ò. å.ïðîáëåìà ïåðåõîäèò â ðàçðÿä òðóäíîðåøàåìûõ çàäà÷. Ëþáûì ÷èñëåí-íûì ìåòîäîì òàêîãî êëàññà çàäà÷è ðåøèòü íåëüçÿ. Â êà÷åñòâå ïðèìåðàíåêîððåêòíîé ïðîáëåìû ðàññìîòðåí êëàññ ðåøåíèé òåîðèè óïðóãîñòè,ïðèâîäÿùèé ê áåñêîíå÷íûì íàïðÿæåíèÿì â óãëîâûõ òî÷êàõ îáëàñòåé èñâÿçàííûõ ñ îáúÿñíåíèåì ôåíîìåíà ðàçðóøåíèÿ â ìåõàíèêå ÷åðåç êî-ýôôèöèåíòû èíòåíñèâíîñòè íàïðÿæåíèé.

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ÀÑÈÌÏÒÎÒÈ×ÅÑÊÈÅ ÑÂÎÉÑÒÂÀ ÐÅØÅÍÈÉÍÅÎÄÍÎÐÎÄÍÎÃÎ ÓÐÀÂÍÅÍÈß

ÂÍÓÒÐÅÍÍÈÕ ÂÎËÍ

Íèêèòåíêî Å.Â.

Ðóáöîâñêèé èíäóñòðèàëüíûé èíñòèòóò (ôèëèàë) Àëòàéñêîãîãîñóäàðñòâåííîãî òåõíè÷åñêîãî óíèâåðñèòåòà èì. È.È. Ïîëçóíîâà,

Ðóáöîâñê, Ðîññèÿ; [email protected]

Â äîêëàäå ïðåäïîëàãàåòñÿ èçëîæèòü íîâûå ðåçóëüòàòû ïî àñèìïòî-òè÷åñêèì ñâîéñòâàì ðåøåíèé çàäà÷è Êîøè äëÿ íåîäíîðîäíîãî óðàâíå-íèÿ âíóòðåííèõ âîëí

∆utt + ux1x1+ ux2x2

= eiλtf(x), t > 0, x ∈ R3,

u|t=0 = 0, ut|t=0 = 0,

ãäå f(x) ∈ S(R3), λ ≥ 0 ïàðàìåòð. Ðåøåíèå äàííîé çàäà÷è îäíî-çíà÷íî îïðåäåëÿåòñÿ â êëàññå ôóíêöèé, óáûâàþùèõ ïðè |x| → ∞ [1].Ïðè âûâîäå àñèìïòîòè÷åñêèõ ðàçëîæåíèé ïðè t→∞ ðåøåíèé u(t, x, λ)èñïîëüçóåòñÿ âàðèàíò ìåòîäà ñòàöèîíàðíîé ôàçû, èçëîæåííîãî â [2].

Â íàñòîÿùåå âðåìÿ èìååòñÿ áîëüøîå ÷èñëî ðàáîò, â êîòîðûõ ïðî-âîäèëèñü èññëåäîâàíèÿ ïîâåäåíèÿ àñèìïòîòèêè ïðè t → ∞ ðåøåíèéêðàåâûõ çàäà÷ äëÿ êîíêðåòíûõ óðàâíåíèé è ñèñòåì, íå ðàçðåøåííûõîòíîñèòåëüíî ñòàðøåé ïðîèçâîäíîé. Ñðåäè íèõ îòìåòèì ðàáîòó [3], âêîòîðîé ðàññìàòðèâàëàñü çàäà÷à Êîøè äëÿ íåîäíîðîäíîãî óðàâíåíèÿÑîáîëåâà

∆utt + uxnxn = eiλtf(x), t > 0, x ∈ Rn,

u|t=0 = 0, ut|t=0 = 0,

ãäå f(x) ∈ S(Rn), λ ≥ 0 ïàðàìåòð, n ≥ 3. Â äàííîé ðàáîòå áûëèóñòàíîâëåíû àñèìïòîòè÷åñêèå ðàçëîæåíèÿ ïðè t→∞ ðåøåíèé u(t, x, λ)â çàâèñèìîñòè îò çíà÷åíèé ïàðàìåòðà λ.


1. Ñåêåðæ-Çåíüêîâè÷ Ñ.ß. Òåîðåìà åäèíñòâåííîñòè è ÿâíîå ïðåäñòàâëåíèåðåøåíèÿ çàäà÷è Êîøè äëÿ óðàâíåíèÿ âíóòðåííèõ âîëí // ÄÀÍ ÑÑÑÐ.1981. Ò. 256, 2. Ñ. 320324.

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2. Óñïåíñêèé Ñ.Â., Äåìèäåíêî Ã.Â., Ïåðåïåëêèí Â. Ã. Òåîðåìû âëîæåíèÿè ïðèëîæåíèÿ ê äèôôåðåíöèàëüíûì óðàâíåíèÿì. Íîâîñèáèðñê: Íàóêà,1984.

3. Áóëäûãåðîâà Ë.Í., Äåìèäåíêî Ã.Â. Àñèìïòîòè÷åñêèå ñâîéñòâà ðåøåíèéíåîäíîðîäíîãî óðàâíåíèÿ Ñîáîëåâà // Íåêëàññè÷åñêèå óðàâíåíèÿ ìàòåìà-òè÷åñêîé ôèçèêè. Íîâîñèáèðñê: Èçä-âî Èí-òà ìàòåìàòèêè ÑÎ ÐÀÍ, 2005.Ñ. 5059.

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ÎÁÎÁÙÅÍÍÎÅ ÐÅØÅÍÈÅÄÈÔÔÅÐÅÍÖÈÀËÜÍÎ-ÎÏÅÐÀÒÎÐÍÎÃÎÓÐÀÂÍÅÍÈß Ñ ÎÒÊËÎÍßÞÙÈÌÑß

ÀÐÃÓÌÅÍÒÎÌ

Îðëîâ Ñ.Ñ.1, Øåìåòîâà Â.Â.2

Èðêóòñêèé ãîñóäàðñòâåííûé óíèâåðñèòåò, Èðêóòñê, Ðîññèÿ;[email protected], [email protected]

Ïóñòü E âåùåñòâåííîå áàíàõîâî ïðîñòðàíñòâî, u è f íåèçâåñòíàÿè çàäàííàÿ ôóíêöèè íåîòðèöàòåëüíîãî äåéñòâèòåëüíîãî àðãóìåíòà t ñîçíà÷åíèÿìè â E. Ðàññìîòðèì óðàâíåíèå

u′(t) = Au(t) +Bu(t− h) + f(t), t > 0, (1)

ãäå A, B : E → E ëèíåéíûå íåïðåðûâíûå îïåðàòîðû, h çàäàííîåïîëîæèòåëüíîå ÷èñëî. Ïî àíàëîãèè ñî ñêàëÿðíûì ñëó÷àåì (E = R) äëÿóðàâíåíèÿ (1) ïîñòàâèì íà÷àëüíóþ çàäà÷ó

u(t) = ϕ(t), −h ≤ t ≤ 0, (2)

ñ çàäàííîé ôóíêöèåé ϕ ∈ C([−h, 0] ; E). Êëàññè÷åñêèì ðåøåíèåì ýòîéçàäà÷è íàçîâåì ôóíêöèþ u ∈ C(t ≥ −h; E) ∩ C1(t > 0; E), êîòîðàÿóäîâëåòâîðÿåò óðàâíåíèþ (1) è íà÷àëüíîìó óñëîâèþ (2).

Ïðîáëåìà ðàçðåøèìîñòè çàäà÷è (1), (2) èçó÷àåòñÿ íà îñíîâå òåîðèèîáîáùåííûõ ôóíêöèé Ñîáîëåâà Øâàðöà ñî çíà÷åíèÿìè â áàíàõîâîìïðîñòðàíñòâå E. Èñïîëüçóåòñÿ êîíöåïöèÿ ôóíäàìåíòàëüíîãî ðåøåíèÿ(ôóíäàìåíòàëüíîé îïåðàòîð-ôóíêöèè [1]) äèôôåðåíöèàëüíîãî îïåðà-òîðà ñ îòêëîíÿþùèìñÿ àðãóìåíòîì. Ýòîò ïîäõîä ïîçâîëÿåò äîêàçàòüñóùåñòâîâàíèå è åäèíñòâåííîñòü ðåøåíèÿ ðàññìàòðèâàåìîé çàäà÷è âêëàññå ðàñïðåäåëåíèé ñ îãðàíè÷åííûì ñëåâà íîñèòåëåì è äàåò ñïîñîáïîñòðîåíèÿ îáîáùåííîãî ðåøåíèÿ, à òàêæå ïîçâîëÿåò íàéòè óñëîâèÿ,ïðè êîòîðûõ îáîáùåííîå è êëàññè÷åñêîå ðåøåíèÿ ñîâïàäàþò.




1. Sidorov N., Loginov B., Sinitsyn A., Falaleev M. LyapunovSchmidt methodsin nonlinear analysis and applications. Dordrecht: Kluwer Acad. Publ., 2002.

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ÈÑÑËÅÄÎÂÀÍÈÅÌÀÒÅÌÀÒÈ×ÅÑÊÈÕ ÌÎÄÅËÅÉ

ÌÈÊÐÎÝËÅÊÒÐÎÌÅÕÀÍÈ×ÅÑÊÎÃÎÐÅÇÎÍÀÒÎÐÀ ÒÈÏÀ ÏËÀÒÔÎÐÌÀ

Ïèìàíîâ Ä.Î.1, Ôàäååâ Ñ.È.1,2, Êîñöîâ Ý. Ã.3

1Íîâîñèáèðñêèé ãîñóäàðñòâåííûé óíèâåðñèòåò,Íîâîñèáèðñê, Ðîññèÿ; [email protected]

2Èíñòèòóò ìàòåìàòèêè èì. Ñ.Ë. Ñîáîëåâà ÑÎ ÐÀÍ,Íîâîñèáèðñê, Ðîññèÿ; [email protected]

3Èíñòèòóò àâòîìàòèêè è ýëåêòðîìåòðèè ÑÎ ÐÀÍ,Íîâîñèáèðñê, Ðîññèÿ; [email protected]

Ïðèâîäÿòñÿ ðåçóëüòàòû ÷èñëåííîãî èññëåäîâàíèÿ ìàòåìàòè÷åñêîéìîäåëè ìèêðîýëåêòðîìåõàíè÷åñêîãî âûñîêî÷àñòîòíîãî ðåçîíàòîðà òè-ïà ïëàòôîðìà êëàññà ÌÝÌÑ (ìèêðîýëåêòðîìåõàíè÷åñêèå ñèñòåìû).Ìàòåìàòè÷åñêèå ìîäåëè ïðèáîðà ïðåäñòàâëåíû íåëèíåéíûìè íà÷àëüíî-êðàåâûìè çàäà÷àìè, îïèñûâàþùèìè êîëåáàíèÿ ïîäâèæíîãî ýëåêòðîäàâ âèäå íåäåôîðìèðóåìîé ïëàòôîðìû, ïðèñîåäèí¼ííîé ê ïðóæèíå ñ öè-ëèíäðè÷åñêîé ôîðìîé èçãèáà. Â êà÷åñòâå ïðóæèíû èñïîëüçóþòñÿ óïðó-ãàÿ áàëêà ñ æ¼ñòêî çàêðåïë¼ííûìè êîíöàìè, óïðóãàÿ áàëêà êîíñîëüíîãîòèïà èëè íàòÿíóòàÿ ïë¼íêà. Ïðè çàïóñêå ðåçîíàòîðà ïîä âîçäåéñòâè-åì ýëåêòðîñòàòè÷åñêîãî ïðèòÿæåíèÿ ìåæäó ïëàòôîðìîé è íåïîäâèæ-íûì ýëåêòðîäîì, ïîêðûòûì ñëîåì äèýëåêòðèêà, âîçíèêàþò êîëåáàíèÿïëàòôîðìû, êîòîðûå çàòåì ïðîäîëæàþòñÿ â âèäå âûñîêî÷àñòîòíûõ ñîá-ñòâåííûõ êîëåáàíèé.

×èñëåííî îïðåäåëåíû óñëîâèÿ âîçíèêíîâåíèÿ êîëåáàíèé, îêàçàâøè-åñÿ, êàê ïîêàçàëî èññëåäîâàíèå ìîäåëè, áëèçêèìè ê ñëó÷àþ, êîãäà ìàñ-ñà ïëàòôîðìû ìíîãî áîëüøå ìàññû ïðóæèíû. Ýòî äàëî âîçìîæíîñòüïîëó÷èòü â àíàëèòè÷åñêîì âèäå ôîðìóëû, äîñòàòî÷íî õîðîøî îïèñû-âàþùèå îñíîâíûå îñîáåííîñòè ðàáîòû ðåçîíàòîðà. Ïðèâåä¼í ïðèìåððàñ÷¼òà ïàðàìåòðîâ ìèêðîðåçîíàòîðà, îïðåäåëÿþùèõ çàäàííûé ðåæèìðàáîòû, êîòîðûé ïîêàçûâàåò, ÷òî íàéäåííûå â ðàìêàõ ìàòåìàòè÷åñêîéìîäåëè çíà÷åíèÿ ïàðàìåòðîâ õàðàêòåðíû äëÿ ÌÝÌÑ.


1. Ôàäååâ Ñ.È., Êîñöîâ Ý. Ã., Ïèìàíîâ Ä.Î. Èññëåäîâàíèå ìàòåìàòè÷åñêîéìîäåëè ìèêðîýëåêòðîìåõàíè÷åñêîãî ðåçîíàòîðà òèïà ïëàòôîðìà // Âû-÷èñëèòåëüíûå òåõíîëîãèè. 2016. Ò. 21, 2. Ñ. 6387.

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ÇÀÄÀ×È ÎÏÒÈÌÀËÜÍÎÃÎ ÓÏÐÀÂËÅÍÈßÄËß ÑÈÑÒÅÌ ÄÐÎÁÍÎÃÎ ÏÎÐßÄÊÀ

Ñ ÐÀÑÏÐÅÄÅËÍÍÛÌÈ ÏÀÐÀÌÅÒÐÀÌÈ

Ïîñòíîâ Ñ.Ñ.

Èíñòèòóò ïðîáëåì óïðàâëåíèÿ èì. Â.À. Òðàïåçíèêîâà ÐÀÍ,Ìîñêâà, Ðîññèÿ; [email protected]

Â ðàáîòå èññëåäîâàíû çàäà÷è îïòèìàëüíîãî óïðàâëåíèÿ ëèíåéíûìèäèíàìè÷åñêèìè ñèñòåìàìè äðîáíîãî ïîðÿäêà ñ ðàñïðåäåë¼ííûìè ïà-ðàìåòðàìè íà ïðèìåðå ñèñòåì, îïèñûâàåìûõ óðàâíåíèåì äèôôóçèè ñäðîáíîé ïðîèçâîäíîé ïî âðåìåíè.

Ðàññìàòðèâàåòñÿ ñèñòåìà ñëåäóþùåãî âèäà:

C0 D

αt Q(x, t) = K

∂2Q(x, t)

∂x2+ u(x, t), (x, t) ∈ Ω = [0,∞)× [0, L],

ãäå Q(x, t) ñîñòîÿíèå, u(x, t) ∈ Lp(Ω) ðàñïðåäåë¼ííîå óïðàâëåíèå,1 < p < ∞, C0 D

αt îïåðàòîð äðîáíîãî äèôôåðåíöèðîâàíèÿ â ñìûñ-

ëå Êàïóòî [1], α ∈ (0, 1], K êîýôôèöèåíò. Íà÷àëüíîå è ãðàíè÷íûåóñëîâèÿ äëÿ äàííîé ñèñòåìû çàäàþòñÿ â âèäå:

Q(x, 0+) = Q0(x), x ∈ [0, L],[b1,2

∂Q(x, t)

∂x+ a1,2Q(x, t)

]x=0,L

= h1,2(t) + u1,2(t), t ≥ 0,

ãäå ai è bi, i = 1, 2, êîýôôèöèåíòû, u1,2(t) ∈ Lp[0, T ] ãðàíè÷íûåóïðàâëåíèÿ. Æåëàåìîå êîíå÷íîå ñîñòîÿíèå îïðåäåëÿåòñÿ óñëîâèåì:

Q(x, T ) = Q∗(x), T > 0, x ∈ [0, L].

Â ðàáîòå èññëåäóþòñÿ äâå ðàçíîâèäíîñòè çàäà÷è îïòèìàëüíîãîóïðàâëåíèÿ: ïîèñê óïðàâëåíèÿ ñ ìèíèìàëüíîé íîðìîé ïðè çàäàííîìâðåìåíè óïðàâëåíèÿ è ïîèñê óïðàâëåíèÿ, îïòèìàëüíîãî ïî áûñòðîäåé-ñòâèþ, ïðè çàäàííîì îãðàíè÷åíèè (â âèäå íåðàâåíñòâà) íà íîðìó óïðàâ-ëåíèÿ. Îáå çàäà÷è ñâîäÿòñÿ ê áåñêîíå÷íîìåðíîé l-ïðîáëåìå ìîìåíòîâ,äëÿ êîòîðîé ñòðîèòñÿ êîíå÷íîìåðíàÿ àïïðîêñèìàöèÿ.

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1. Kilbas A.A., Srivastava H.M., Trujillo J. J. Theory and applications of frac-tional dierential equations. Amsterdam: Elsevier, 2006.

2. Êóáûøêèí Â.À., Ïîñòíîâ Ñ.Ñ. Çàäà÷à îïòèìàëüíîãî óïðàâëåíèÿ äëÿ ëè-íåéíûõ ðàñïðåäåë¼ííûõ ñèñòåì äðîáíîãî ïîðÿäêà // Âåñòí. ÐÓÄÍ. Ñåð.Ìàòåìàòèêà, ôèçèêà, èíôîðìàòèêà. 2014. 2. Ñ. 381385.

3. Kubyshkin V.A., Postnov S. S. The optimal control problem for linear systemsof non-integer order with lumped and distributed parameters // Discontin.Nonlinearity Complex. 2015. V. 4, No 4. P. 429443.

4. Kubyshkin V.A., Postnov S. S. Optimal boundary controls for the systemsdescribed by diusion-type equation with fractional-order time derivative //Proc. Int. Conf. Stability and Oscillations of Nonlinear Control Systems(Pyatnitskiy's Conference). Moscow: IEEE, 2016. P. 168170.

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ÒÎ×ÍÛÅ ÒÅÎÐÅÌÛ Î ÏÐÎÄÎËÆÅÍÈßÕÑ ÌÈÍÈÌÀËÜÍÎÉ ÍÎÐÌÎÉ

Ðàìàçàíîâ Ì.Ä.

Èíñòèòóò ìàòåìàòèêè ñ âû÷èñëèòåëüíûì öåíòðîì ÓÍÖ ÐÀÍ,Óôà, Ðîññèÿ; [email protected]

Ïóñòü B ÿâëÿåòñÿ áàíàõîâûì ïðîñòðàíñòâîì îïðåäåëåííûõ íà Rnôóíêöèé. Òðåáóåòñÿ çàäàííóþ íà ïðîèçâîëüíîì ìíîæåñòâå ω ⊂ Rnôóíêöèþ ïðîäîëæèòü íà âñå ïðîñòðàíñòâî ñ ìèíèìàëüíîé íîðìîé. Ïî-ëàãàÿ, ÷òî òàêîå ïðîäîëæåíèå âîçìîæíî, ìû äàåì åãî ôîðìóëó.

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ÎÁÐÀÒÍÛÅ ÇÀÄÀ×È ÏÎ ÎÏÐÅÄÅËÅÍÈÞÏÐÀÂÛÕ ×ÀÑÒÅÉ ÓÐÀÂÍÅÍÈÉ

ÑÌÅØÀÍÍÎÃÎ ÒÈÏÀ

Ñàáèòîâ Ê.Á.1, Ñèäîðîâ Ñ.Í.2

Ñòåðëèòàìàêñêèé ôèëèàë Áàøêèðñêîãî ãîñóäàðñòâåííîãîóíèâåðñèòåòà, Ñòåðëèòàìàêñêèé ôèëèàë Èíñòèòóòàñòðàòåãè÷åñêèõ èññëåäîâàíèé, Ñòåðëèòàìàê, Ðîññèÿ;


Ðàññìîòðèì óðàâíåíèå ñìåøàííîãî òèïà

Ln,mu = F (x, t), (1)

çäåñü

Ln,mu =

tnuxx − ut − b2tnu, t > 0,

(−t)muxx − utt − b2(−t)mu, t < 0,

F (x, t) =

f1(x)g1(t), t > 0,

f2(x)g2(t), t < 0,

â ïðÿìîóãîëüíîé îáëàñòè D = (x, t) : 0 < x < l, −α < t < β, ãäån ≥ 0, m ≥ 0, b ≥ 0, l > 0, α > 0, β > 0 çàäàííûå äåéñòâèòåëüíûå÷èñëà, è ñëåäóþùèå îáðàòíûå çàäà÷è.

Çàäà÷à 1. Íàéòè ôóíêöèè u(x, t) è g1(t), óäîâëåòâîðÿþùèå óñëî-âèÿì:

u(x, t) ∈ C(D) ∩ C1t (D) ∩ C1

x(D) ∩ C2x(D+) ∩ C2(D−); (2)

g1(t) ∈ C[0, β]; (3)

Ln,mu(x, t) ≡ F (x, t), (x, t) ∈ D+ ∪D−; (4)

u(0, t) = u(l, t) = 0, −α ≤ t ≤ β; u(x,−α) = 0, 0 ≤ x ≤ l; (5)

u(x0, t) = h1(t), 0 < x0 < l, 0 ≤ t ≤ β, (6)

ãäå fi(x), i = 1, 2, g2(t), h1(t) çàäàííûå ôóíêöèè, x0 çàäàííàÿ òî÷êàèç èíòåðâàëà (0, l), D+ = D ∩ t > 0, D− = D ∩ t < 0.

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Çàäà÷à 2. Íàéòè ôóíêöèè u(x, t) è g2(t), óäîâëåòâîðÿþùèå óñëîâè-ÿì (2), (4), (5) è

g2(t) ∈ C[−α, 0]; u(x0, t) = h2(t), 0 < x0 < l, −α ≤ t ≤ 0, (7)

ãäå fi(x), i = 1, 2, g1(t), h2(t) èçâåñòíûå ôóíêöèè.Çàäà÷à 3. Íàéòè ôóíêöèè u(x, t), g1(t), g2(t), óäîâëåòâîðÿþùèå

óñëîâèÿì (2)(7), çäåñü fi(x), hi(t), i = 1, 2, çàäàííûå ôóíêöèè.Îòìåòèì, ÷òî èññëåäîâàíèå çàäà÷ 13 áàçèðóåòñÿ íà ïðÿìîé íà÷àëü-

íî-ãðàíè÷íîé çàäà÷å (2), (4), (5), èçó÷åííîé â ðàáîòå [1]. Ïîñòàâëåííûåçàäà÷è äëÿ óðàâíåíèÿ (1) ïðè n = m = 0 èçó÷åíû â [2, ñ. 228238].

Â äàííîé ðàáîòå äîêàçàíû ñîîòâåòñòâóþùèå òåîðåìû åäèíñòâåííî-ñòè è ñóùåñòâîâàíèÿ ðåøåíèé çàäà÷ 13 äëÿ óðàâíåíèÿ (1) ïðè n = 0,m > 0.


1. Ñàáèòîâ Ê.Á., Ñèäîðîâ Ñ.Í. Íà÷àëüíî-ãðàíè÷íàÿ çàäà÷à äëÿ íåîäíîðîä-íûõ âûðîæäàþùèõñÿ óðàâíåíèé ñìåøàííîãî ïàðàáîëî-ãèïåðáîëè÷åñêîãîòèïà // Èòîãè íàóêè è òåõíèêè. Ñåð. Ñîâðåì. ìàò. è åå ïðèë. Òåìàò. îáç.2017. Ò. 137. Ñ. 2660.

2. Ñàáèòîâ Ê.Á. Ïðÿìûå è îáðàòíûå çàäà÷è äëÿ óðàâíåíèé ñìåøàííîãîïàðàáîëî-ãèïåðáîëè÷åñêîãî òèïà. Ì.: Íàóêà, 2016.

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ÎÁ ÎÖÅÍÊÀÕ ÐÅØÅÍÈÉÍÅÀÂÒÎÍÎÌÍÎÃÎ ËÈÍÅÉÍÎÃÎÓÐÀÂÍÅÍÈß Ñ ÇÀÏÀÇÄÛÂÀÍÈÅÌ

Ñåäîâà Í.Î.

Óëüÿíîâñêèé ãîñóäàðñòâåííûé óíèâåðñèòåò, Óëüÿíîâñê, Ðîññèÿ;[email protected]

Äëÿ ñèñòåì ñ çàïàçäûâàíèåì äàæå â ëèíåéíîì ñëó÷àå êðèòåðèèóñòîé÷èâîñòè èçâåñòíû ëèøü äëÿ î÷åíü íåáîëüøîãî êëàññà ñêàëÿðíûõóðàâíåíèé; â îáùåì ñëó÷àå ðàçðàáîòàíû ëèøü ñïîñîáû ïîëó÷åíèÿ äîñ-òàòî÷íûõ óñëîâèé óñòîé÷èâîñòè è ïîñòðîåíèÿ îöåíîê äëÿ ðåøåíèéíåêîòîðûõ òèïîâ ñèñòåì. Ïðîâåðêà èìåþùèõñÿ óñëîâèé, â ñâîþ î÷åðåäü,ìîæåò îêàçàòüñÿ äîñòàòî÷íî ñåðüåçíîé ñàìîñòîÿòåëüíîé çàäà÷åé äàæåäëÿ ëèíåéíîé ñèñòåìû ñ ïîñòîÿííûìè ïàðàìåòðàìè, à ïðåèìóùåñòâàòîãî èëè èíîãî ñïîñîáà çàâèñÿò îò êîíêðåòíîãî âèäà ñèñòåìû. Ïîýòîìóóíèâåðñàëüíûõ ðåöåïòîâ â ýòîé îáëàñòè ïî ñåé äåíü íå ñóùåñòâóåò, èèññëåäîâàíèÿ â ýòîì íàïðàâëåíèè ïðîäîëæàþòñÿ.

Ïðè ýòîì ìíîãèå èçâåñòíûå ðåçóëüòàòû îïèðàþòñÿ íà ñâîéñòâà ñêà-ëÿðíîãî óðàâíåíèÿ âèäà

x(t) = −a(t)x(t− r(t)), (1)

êîòîðîìó ïîñâÿùåíû ìíîãî÷èñëåííûå èññëåäîâàíèÿ, íà÷èíàÿ ñ ðàáîòÀ.Ä. Ìûøêèñà è Ê. Êóêà [1, 2]. Çíà÷èòåëüíàÿ äîëÿ ïîëó÷åííûõ ê íàñòî-ÿùåìó âðåìåíè ðåçóëüòàòîâ îá îöåíêàõ ðåøåíèé óðàâíåíèÿ (1) èñïîëü-çóåò â òîì èëè èíîì âèäå òàê íàçûâàåìûå 3/2-ïðèçíàêè (íåêîòîðûåññûëêè íà ñîîòâåòñòâóþùèå èñòî÷íèêè ïðåäñòàâëåíû â ñòàòüå [3]). Âôîðìå òàêèõ ïðèçíàêîâ ïðåäñòàâëåíû è òî÷íûå (íåóëó÷øàåìûå â îá-ùåì ñëó÷àå) óñëîâèÿ óñòîé÷èâîñòè è îãðàíè÷åííîñòè ðåøåíèé äëÿ óðàâ-íåíèÿ (1), êîòîðûå, îäíàêî, ìîãóò áûòü ñóùåñòâåííî îñëàáëåíû ïðè íà-ëè÷èè íåêîòîðûõ ñïåöèàëüíûõ ñâîéñòâ ôóíêöèé a(t) è r(t).

Â äîêëàäå ïðåäëàãàåòñÿ îáçîð è ñðàâíåíèå ðåçóëüòàòîâ îá óñòîé÷è-âîñòè è îöåíêàõ ðåøåíèé äëÿ óðàâíåíèÿ (1), à òàêæå îáñóæäàþòñÿ âîç-ìîæíîñòè èõ ìîäèôèêàöèè ïðè ðàçëè÷íûõ äîïîëíèòåëüíûõ óñëîâèÿõ.Äëÿ îáîñíîâàíèÿ èñïîëüçóþòñÿ íåêëàññè÷åñêèå ôóíêöèè Ëÿïóíîâà.Ðàññìàòðèâàþòñÿ, â òîì ÷èñëå, ñïîñîáû ïîëó÷åíèÿ îöåíîê ðåøåíèé íàîñíîâå èñïîëüçîâàíèÿ ôóíêöèé, ïðîèçâîäíàÿ êîòîðûõ ïðè óñëîâèÿõ Ðà-çóìèõèíà ìîæåò áûòü çíàêîïåðåìåííîé (ñì., íàïðèìåð, [4]). Ïðè ýòîì

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äîïóñêàþòñÿ, âîîáùå ãîâîðÿ, êàê îãðàíè÷åííûå, òàê è íåîãðàíè÷åííûåôóíêöèè r(t) ≥ 0 (ïðè óñëîâèè, ÷òî t − r(t) íåîãðàíè÷åííî âîçðàñòàåòïðè t → +∞) â ïðåäïîëîæåíèè, ïî êðàéíåé ìåðå, ëîêàëüíîé èíòåãðè-ðóåìîñòè êîýôôèöèåíòà a(t). Ïîëó÷åíû òàêæå ðåçóëüòàòû îá óñòîé÷è-âîñòè äëÿ íåêîòîðûõ âèäîâ ëèíåéíûõ ñèñòåì ñ çàïàçäûâàíèåì ñ ïåðå-ìåííûìè ïàðàìåòðàìè.


1. Ìûøêèñ À.Ä. Î ðåøåíèÿõ ëèíåéíûõ îäíîðîäíûõ äèôôåðåíöèàëüíûõóðàâíåíèé ïåðâîãî ïîðÿäêà óñòîé÷èâîãî òèïà ñ çàïàçäûâàþùèì àðãóìåí-òîì // Ìàò. ñá. 1951. Ò. 28, 3. Ñ. 641658.

2. Cooke K. L. Asymptotic theory for the delay-dierential equation u′(t) =−au(t− r(u(t))) // J. Math. Anal. Appl. 1967. V. 19. P. 160173.

3. Knyazhishche L.B., Shcheglov V.A. On the sign deniteness of Liapunovfunctionals and stability of a linear delay equation // Electron. J. Qual. TheoryDier. Equ. 1998. No 8. P. 113.

4. Zhou B., Egorov A.V. Razumikhin and Krasovskii stability theorems for time-varying time-delay systems // Automatica. 2016. V. 71. P. 281291.

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ÌÀÒÅÌÀÒÈ×ÅÑÊÎÅ ÌÎÄÅËÈÐÎÂÀÍÈÅÃÎÐÌÎÍÀËÜÍÎÃÎ ËÅ×ÅÍÈß

ÎÍÊÎËÎÃÈ×ÅÑÊÈÕ ÇÀÁÎËÅÂÀÍÈÉÂ ÓÑËÎÂÈßÕ ÃÎÐÌÎÍÎÐÅÇÈÑÒÅÍÒÍÎÑÒÈ

Ñåðîâàéñêèé Ñ.ß.1, Íóðñåèòîâ Ä.Á.2, Àçèìîâ À.À.3

1Êàçàõñêèé íàöèîíàëüíûé óíèâåðñèòåò èì. àëü-Ôàðàáè,Àëìàòû, Ðåñïóáëèêà Êàçàõñòàí; [email protected]

2Êàçàõñêèé íàöèîíàëüíûé èññëåäîâàòåëüñêèé ïîëèòåõíè÷åñêèéóíèâåðñèòåò èì. Ê. Ñàòïàåâà, Àëìàòû, Ðåñïóáëèêà Êàçàõñòàí;

[email protected]Êàçàõñêèé íàöèîíàëüíûé èññëåäîâàòåëüñêèé ïîëèòåõíè÷åñêèéóíèâåðñèòåò èì. Ê. Ñàòïàåâà, Àëìàòû, Ðåñïóáëèêà Êàçàõñòàí;

[email protected]

Äëÿ ëå÷åíèÿ íåêîòîðûõ ôîðì ðàêà, ñâÿçàííûõ ñ äåéñòâèåì ãîðìî-íîâ, ïðèìåíÿåòñÿ ãîðìîíàëüíîå ëå÷åíèå. Ïîäàâëÿÿ âûðàáîòêó îðãàíèç-ìîì ãîðìîíîâ, îíî ïîçâîëÿåò çàòîðìîçèòü ðàçâèòèå îïóõîëè. Îäíàêîäåéñòâèå ãîðìîíàëüíîãî ëå÷åíèÿ íîñèò âðåìåííûé õàðàêòåð ââèäó ïî-ñòåïåííîãî ïðèâûêàíèÿ êëåòîê ðàêà ê äåéñòâèþ ãîðìîíîâ, ò. å. ÿâëåíèÿãîðìîíîðåçèñòåíòíîñòè. Ýòî îáúÿñíÿåòñÿ òåì, ÷òî îïóõîëü ñîñòîèò èçäâóõ òèïîâ êëåòîê ÷óâñòâèòåëüíûõ è ðåçèñòåíòíûõ ê äåéñòâèþ ãîð-ìîíàëüíûõ ïðåïàðàòîâ. Â óñëîâèÿõ ãîðìîíàëüíîãî ëå÷åíèÿ ïðîèñõîäèòïîñòåïåííîå çàìåùåíèå ÷óâñòâèòåëüíûõ ðàêîâûõ êëåòîê ðåçèñòåíòíû-ìè, ÷òî ïðèâîäèò ê äàëüíåéøåìó ðàçâèòèþ îïóõîëè. Ïðåäëàãàåòñÿ ìà-òåìàòè÷åñêàÿ ìîäåëü ïðîöåññà, áàçèðóþùàÿñÿ íà îäíîôàçíîé çàäà÷åÑòåôàíà.

Ïðè âûâîäå ìîäåëè èñïîëüçóþòñÿ ñëåäóþùèå ïðåäïîëîæåíèÿ. Îïó-õîëü îáëàäàåò ñôåðè÷åñêîé ñèììåòðèåé è ñîñòîèò èç äâóõ òèïîâ ðàêî-âûõ êëåòîê, îäíè èç êîòîðûõ ÿâëÿþòñÿ ÷óâñòâèòåëüíûìè ê äåéñòâèþãîðìîíîâ, à äðóãèå ÿâëÿþòñÿ ðåçèñòåíòíûìè. Â îòñóòñòâèè ëå÷åíèÿ íà-áëþäàåòñÿ ðîñò îïóõîëè â ñîîòâåòñòâèè ñ ìîäåëüþ Ôåðõþëüñòà. Ðàñïðî-ñòðàíåíèå ðàêîâûõ êëåòîê âíóòðè îïóõîëè èìååò äèôôóçèîííóþ ïðèðî-äó. Ãîðìîíàëüíîå ëå÷åíèå ïîäàâëÿåò ðîñò ÷óâñòâèòåëüíûõ ê ãîðìîíàìðàêîâûõ êëåòîê. Ðàêîâûå êëåòêè ìîãóò ïðèîáðåòàòü è òåðÿòü ãîðìîíî-ðåçèñòåíòíîñòü. Â ðåçóëüòàòå ïîëó÷àåì ñëåäóþùèå óðàâíåíèÿ

∂us∂t

= Ds1

ρ2

∂

∂ρ

(ρ2 ∂us

∂ρ

)+[ as

1 + f(t)uθs− bs(us + ur)

]us + crsur,

93


ρ0 < ρ < ξ(t), t > 0,

∂ur∂t

= Dr1

ρ2

∂

∂ρ

(ρ2 ∂ur

∂ρ

)+[ar − br(us + ur)

]ur + csrus,

ρ0 < ρ < ξ(t), t > 0,

ñ îïðåäåëåííûìè íà÷àëüíûìè óñëîâèÿìè, óñëîâèåì Íåéìàíà íà ëåâîéãðàíèöå è óñëîâèÿìè Ñòåôàíà íà ïðàâîé ãðàíèöå, ãäå us êîíöåíòðà-öèÿ ÷óâñòâèòåëüíûõ ðàêîâûõ êëåòîê, ur êîíöåíòðàöèÿ ðåçèñòåíòíûõðàêîâûõ êëåòîê, ξ = ξ(t) ãðàíèöà îïóõîëè.

Ðàñ÷åòû ïîêàçûâàþò, ÷òî â îòñóòñòâèè ëå÷åíèÿ, ÷òî ñîîòâåòñòâóåòñëó÷àþ ðàâåíñòâà íóëþ êîíöåíòðàöèè f(t) ëåêàðñòâåííûõ ïðåïàðàòîâ,ôóíêöèÿ ξ, õàðàêòåðèçóþùàÿ ðàçìåðû îïóõîëè, ðàñòåò. Ñ íà÷àëîì ëå-÷åíèÿ íàáëþäàåòñÿ çíà÷èòåëüíîå óìåíüøåíèå êîíöåíòðàöèè us ÷óâñòâè-òåëüíûõ êëåòîê, ñóùåñòâåííî ïðåîáëàäàþùèõ èçíà÷àëüíî. Âñëåäñòâèåýòîãî ïðîèñõîäèò ñîêðàùåíèå ðàçìåðîâ îïóõîëè. Îäíàêî ïîñòåïåííî íà-÷èíàåòñÿ ðîñò êîíöåíòðàöèè ur ðåçèñòåíòíûõ êëåòîê, íà êîòîðûå ëå÷å-íèå íå îêàçûâàåò âîçäåéñòâèÿ. Òåì ñàìûì ñîêðàùåíèå îïóõîëè ñìåíÿ-åòñÿ åå óâåëè÷åíèåì, à ýôôåêòèâíîñòü ëå÷åíèÿ ïîñòåïåííî ñõîäèò íàíåò.

Óòî÷íåíèå ìàòåìàòè÷åñêîé ìîäåëè îñóùåñòâëÿåòñÿ â ïðîöåññå ðåøå-íèÿ ñîîòâåòñòâóþùèõ îáðàòíûõ çàäà÷, ñâÿçàííûõ ñ âîññòàíîâëåíèåì ååïàðàìåòðîâ ïî ðåçóëüòàòàì èçìåðåíèÿ ñîñòîÿíèÿ ñèñòåìû.

94


ÓÑÒÎÉ×ÈÂÎÑÒÜ ÏÎËÎÆÅÍÈÉ ÐÀÂÍÎÂÅÑÈßÂ ÎÄÍÎÉ ÌÎÄÅËÈ ÇÀÁÎËÅÂÀÍÈß

Ñêâîðöîâà Ì.À.


[email protected]

Ðàññìàòðèâàåòñÿ ñèñòåìà äèôôåðåíöèàëüíûõ óðàâíåíèé ñ çàïàçäû-âàþùèì àðãóìåíòîì, îïèñûâàþùàÿ âçàèìîäåéñòâèå êëåòîê çäîðîâîãîîðãàíèçìà ñ âèðóñàìè [1]:

d

dtx(t) = λ− dx(t)− βx(t)v(t),

d

dty(t) = βx(t− τ)v(t− τ)e−ατ − ay(t),

d

dtv(t) = ky(t)− uv(t),

(1)

ãäå x(t) êîëè÷åñòâî (êîíöåíòðàöèÿ) çäîðîâûõ êëåòîê, y(t) êîëè-÷åñòâî (êîíöåíòðàöèÿ) çàðàæåííûõ êëåòîê, v(t) êîëè÷åñòâî (êîí-öåíòðàöèÿ) âèðóñîâ. Èçó÷àåòñÿ àñèìïòîòè÷åñêîå ïîâåäåíèå ðåøåíèé âîêðåñòíîñòè ïîëîæåíèé ðàâíîâåñèÿ ñèñòåìû (1). Ïîëó÷åíû óñëîâèÿ íàêîýôôèöèåíòû ñèñòåìû, ïðè êîòîðûõ ïîëîæåíèÿ ðàâíîâåñèÿ ÿâëÿþòñÿàñèìïòîòè÷åñêè óñòîé÷èâûìè. Èñïîëüçóÿ ìîäèôèöèðîâàííûå ôóíêöè-îíàëû Ëÿïóíîâà Êðàñîâñêîãî [2], óñòàíîâëåíû îöåíêè ñêîðîñòè ñõî-äèìîñòè ðåøåíèé ê ïîëîæåíèÿì ðàâíîâåñèÿ è îöåíêè íà îáëàñòè ïðè-òÿæåíèÿ.




1. Herz A.V.M., Bonhoeer S., Anderson R.M., May R.M., Nowak M.A. Viraldynamics in vivo: limitations on estimates of intercellular delay and virusdecay // Proc. Natl. Acad. Sci. USA (Medical Sciences).1996. V. 93. P. 72477251.

2. Äåìèäåíêî Ã.Â., Ìàòâååâà È.È. Àñèìïòîòè÷åñêèå ñâîéñòâà ðåøåíèé äèô-ôåðåíöèàëüíûõ óðàâíåíèé ñ çàïàçäûâàþùèì àðãóìåíòîì // Âåñòí. ÍÃÓ.Ñåð. Ìàòåìàòèêà, ìåõàíèêà, èíôîðìàòèêà. 2005. Ò. 5, âûï. 3. Ñ. 2028.

95


ÄÈÔÔÅÐÅÍÖÈÀËÜÍÎ-ÈÍÂÀÐÈÀÍÒÍÛÅÐÅØÅÍÈß È ÊÐÀÒÍÛÅ ÂÎËÍÛ

Òàëûøåâ À.À.


Ðåøåíèå U = (u1, . . . , um) ñèñòåìû äèôôåðåíöèàëüíûõ óðàâíåíèé

n∑α=1

m∑β=1

aαβj(u)∂uβ

∂xα= 0, j = 1, . . . , k, (1)

íàçûâàåòñÿ p-êðàòíîé âîëíîé [1, 23], åñëè

rank∂(u1, . . . , um)

∂(x1, . . . , xn)= p < minn,m.

Ñèñòåìà (1) äîïóñêàåò (n + 1)-ïàðàìåòðè÷åñêóþ ëîêàëüíóþ ãðóï-ïó Ëè Gn+1, ïîðîæäàåìóþ ïðåîáðàçîâàíèÿìè ñäâèãîâ è ðàñòÿæåíèÿíåçàâèñèìûõ ïåðåìåííûõ. Â ðàáîòå [1, 23] îòìå÷àåòñÿ, ÷òî êàæäàÿp-êðàòíàÿ âîëíà ñèñòåìû (1) ÿâëÿåòñÿ ÷àñòè÷íî èíâàðèàíòíûì Gn+1-ðåøåíèåì ðàíãà p.

Ìíîæåñòâî ÷àñòè÷íî èíâàðèàíòíûõ ðåøåíèé ÿâëÿåòñÿ îáúåäèíåíè-åì íåñêîëüêèõ êëàññîâ äèôôåðåíöèàëüíî-èíâàðèàíòíûõ ðåøåíèé [2]. Âíàñòîÿùåé ðàáîòå p-êðàòíûå âîëíû èçó÷àþòñÿ êàê äèôôåðåíöèàëüíî-èíâàðèàíòíûå ðåøåíèÿ ëîêàëüíîé ãðóïïû Ëè Gn+1. Ïðèâîäÿòñÿ ïðè-ìåðû äèôôåðåíöèàëüíî-èíâàðèàíòíûõ Gn+1-ðåøåíèé âîëíîâîãî óðàâ-íåíèÿ.


1. Îâñÿííèêîâ Ë.Â. Ãðóïïîâîé àíàëèç äèôôåðåíöèàëüíûõ óðàâíåíèé. Ì.:Íàóêà, 1978.

2. Òàëûøåâ À.À. Î äèôôåðåíöèàëüíî-èíâàðèàíòíûõ ðåøåíèÿõ // Ñèá.æóðí. ÷èñò. è ïðèêë. ìàòåì. 2016. Ò. 16, âûï. 3. C. 7584.

96


ËÈÍÅÉÍÀß ÎÁÐÀÒÍÀß ÇÀÄÀ×ÀÑ ÃÐÀÍÈ×ÍÛÌ ÏÅÐÅÎÏÐÅÄÅËÅÍÈÅÌ

Â ÌÍÎÃÎÌÅÐÍÎÌ ÑËÓ×ÀÅ

Òåëåøåâà Ë.À.

Áóðÿòñêèé ãîñóäàðñòâåííûé óíèâåðñèòåò, Óëàí-Óäý, Ðîññèÿ;[email protected]

Â öèëèíäðå Q = (x, t) : x ∈ Ω, t ∈ (0, T ), T < ∞ äëÿ óðàâíåíèÿâèäà

ut + ∆2u+ c(x)u = f(x, t) + q(t)h(x, t)

ðàññìîòðåíà îáðàòíàÿ çàäà÷à ñ ãðàíè÷íûì ïåðåîïðåäåëåíèåì â ìíîãî-ìåðíîì ñëó÷àå. Ìåòîäû èññëåäîâàíèÿ ðàññìàòðèâàåìîé çàäà÷è îñíîâà-íû íà ïåðåõîäå îò îáðàòíîé çàäà÷è ê íîâîé, óæå ïðÿìîé, íî ïðè ýòîìíåëîêàëüíîé (ãðàíè÷íî-íåëîêàëüíîé) êðàåâîé çàäà÷å äëÿ íàãðóæåííî-ãî äèôôåðåíöèàëüíîãî óðàâíåíèÿ, äîêàçàòåëüñòâå å¼ ðàçðåøèìîñòè èïîñòðîåíèè ðåøåíèÿ èñõîäíîé îáðàòíîé çàäà÷è. Âîçíèêàþùóþ ïðè ðå-äóêöèè èçó÷àåìîé îáðàòíîé çàäà÷è íåëîêàëüíóþ çàäà÷ó ìîæíî òðàê-òîâàòü êàê îáîáùåíèå îäíîãî èç ñëó÷àåâ íåëîêàëüíîé êðàåâîé çàäà÷èÀ.À. Ñàìàðñêîãî, ïðåäëîæåííîé äëÿ îäíîìåðíîãî óðàâíåíèÿ òåïëîïðî-âîäíîñòè [1], ïðè÷åì îáîáùåíèå êàê ìíîãîìåðíîå, òàê è îáîáùåíèå íàïàðàáîëè÷åñêèå óðàâíåíèÿ âûñîêîãî ïîðÿäêà. Ïðåäñòàâëÿåòñÿ, ÷òî ïî-ëó÷åííûå â íàñòîÿùåé ðàáîòå ðåçóëüòàòû î ðàçðåøèìîñòè ìíîãîìåðíûõàíàëîãîâ çàäà÷è À.À. Ñàìàðñêîãî, à òàêæå ñâÿçàííûõ ñ íèìè êðàåâûõçàäà÷ äëÿ íàãðóæåííûõ óðàâíåíèé èìåþò è ñàìîñòîÿòåëüíîå çíà÷åíèå.

Ïî ïîñòàíîâêàì çàäà÷ è ïî ìåòîäàì ðåøåíèé êàê íàèáîëåå áëèçêèåìîæíî îòìåòèòü ñòàòüè [24].


1. Ñàìàðñêèé À.À. Î íåêîòîðûõ ïðîáëåìàõ òåîðèè äèôôåðåíöèàëüíûõóðàâíåíèé // Äèôôåðåíö. óðàâíåíèÿ. 1980. Ò. 16, 11. Ñ. 19251935.

2. Êîæàíîâ À.È. Î ðàçðåøèìîñòè íåêîòîðûõ íåëîêàëüíûõ è ñâÿçàííûõ ñ íè-ìè îáðàòíûõ çàäà÷ äëÿ ïàðàáîëè÷åñêèõ óðàâíåíèé // Ìàò. çàìåòêè ßÃÓ.2011. Ò. 18, âûï. 2. Ñ. 6478.

3. Òåëåøåâà Ë.À. Î ðàçðåøèìîñòè ëèíåéíîé îáðàòíîé çàäà÷è äëÿ ïàðàáî-ëè÷åñêîãî óðàâíåíèÿ âûñîêîãî ïîðÿäêà // Ìàò. çàìåòêè ßÃÓ. 2013. Ò. 20,âûï. 2. Ñ. 186196.

4. Êîæàíîâ À.È., Òåëåøåâà Ë.À. Ïàðàáîëè÷åñêèå óðàâíåíèÿ âûñîêîãî ïî-ðÿäêà: îáðàòíûå çàäà÷è ñ ãðàíè÷íûì ïåðåîïðåäåëåíèåì è ãðàíè÷íî-íåëîêàëüíûå êðàåâûå çàäà÷è // Äîêë. ÀÌÀÍ. 2015. Ò. 17, 4. Ñ. 4260.

97


ÀÑÈÌÏÒÎÒÈ×ÅÑÊÎÅ ÏÎÂÅÄÅÍÈÅÐÅØÅÍÈÉ ÎÄÍÎÃÎ ÓÐÀÂÍÅÍÈß

Ñ ÇÀÏÀÇÄÛÂÀÞÙÈÌ ÀÐÃÓÌÅÍÒÎÌ

Óâàðîâà È.À.


Â ðàáîòàõ Ã.Â. Äåìèäåíêî è åãî ó÷åíèêîâ (ñì., íàïðèìåð, îáçîðíóþðàáîòó [1]) áûëè óñòàíîâëåíû ñâÿçè ìåæäó ðåøåíèÿìè êëàññîâ ñèñòåìîáûêíîâåííûõ äèôôåðåíöèàëüíûõ óðàâíåíèé áîëüøîé ðàçìåðíîñòè n

dx

dt= Anx+ Fn(t, x) (1)

è ðåøåíèÿìè óðàâíåíèÿ ñ çàïàçäûâàþùèì àðãóìåíòîì

dy(t)

dt= −θy(t) + g(t− τ, y(t− τ)), t > τ. (2)

Èñïîëüçóÿ ïîëó÷åííûå ðàíåå îöåíêè äëÿ ðåøåíèé (1), (2) (ñì. [2, 3]),â äàííîé ðàáîòå ìû èññëåäóåì àñèìïòîòè÷åñêîå ïîâåäåíèå ðåøåíèé óðàâ-íåíèÿ (2) ïðè t→∞.




1. Äåìèäåíêî Ã.Â. Î êëàññàõ ñèñòåì äèôôåðåíöèàëüíûõ óðàâíåíèé âûñîêîéðàçìåðíîñòè è óðàâíåíèÿõ ñ çàïàçäûâàþùèì àðãóìåíòîì // Èòîãè íàóêè.Þã Ðîññèè. Ñåð.: Ìàòåìàòè÷åñêèé ôîðóì. Âëàäèêàâêàç: ÞÌÈ ÂÍÖ ÐÀÍè ÐÑÎ-À, 2011. Ò. 5. Ñ. 4556.

2. Äåìèäåíêî Ã.Â., Ìåëüíèê (Óâàðîâà) È.À. Îá îäíîì ñïîñîáå àïïðîêñèìà-öèè ðåøåíèé äèôôåðåíöèàëüíûõ óðàâíåíèé ñ çàïàçäûâàþùèì àðãóìåí-òîì // Ñèá. ìàò æóðí. 2010. Ò. 51, 3. Ñ. 528546.

3. Óâàðîâà È.À. Ïðåäåëüíûå òåîðåìû íà ïîëóîñè äëÿ îäíîãî êëàññà ñèñòåìîáûêíîâåííûõ äèôôåðåíöèàëüíûõ óðàâíåíèé âûñîêîé ðàçìåðíîñòè //Ìåæäóíàðîäíàÿ êîíôåðåíöèÿ Ìàòåìàòèêà â ñîâðåìåííîì ìèðå. Òåçèñûäîêëàäîâ. Íîâîñèáèðñê: Èçä-âî Èíñòèòóòà ìàòåìàòèêè, 2017. Ñ. 255.

98


ÐÀÇÐÅØÈÌÎÑÒÜ Â ÑËÀÁÎÌ ÑÌÛÑËÅÍÀ×ÀËÜÍÎ-ÊÐÀÅÂÎÉ ÇÀÄÀ×È ÄËß

ÑÈÑÒÅÌÛ ÎÑÊÎËÊÎÂÀ ÏÀÂËÎÂÑÊÎÃÎ

Óñòþæàíèíîâà À.Ñ.1, Òóðáèí Ì.Â.2

Âîðîíåæñêèé ãîñóäàðñòâåííûé óíèâåðñèòåò, Âîðîíåæ, Ðîññèÿ;[email protected], [email protected]

Ïóñòü Ω îãðàíè÷åííàÿ îáëàñòü â Rn (n = 2, 3) ñ ãðàíèöåé ∂Ωêëàññà C3. Íà ïðîìåæóòêå âðåìåíè [0, T ], 0 < T <∞, ðàññìàòðèâàåòñÿñëåäóþùàÿ ñèñòåìà óðàâíåíèé:

∂v

∂t− ν∆v +

n∑i=1

vi∂v

∂xi− κ

∂∆v

∂t− κ

n∑k=1

vk∂∆v

∂xk+∇p = f, div v = 0. (1)

Äëÿ ñèñòåìû (1) ðàññìîòðèì íà÷àëüíî-êðàåâóþ çàäà÷ó ñ íà÷àëüíûìè ãðàíè÷íûì óñëîâèÿìè

v|t=0 = a(x), x ∈ Ω, v|∂Ω×[0,T ] = 0. (2)

Ïóñòü f ∈ L2(0, T ;V −1), a ∈ V 2.Îïðåäåëåíèå. Ñëàáûì ðåøåíèåì íà÷àëüíî-êðàåâîé çàäà÷è (1)(2)

íàçûâàåòñÿ ôóíêöèÿ v ∈ W =u : u ∈ C([0, T ];V 2), u′ ∈ L2(0, T ;V 2)

,

óäîâëåòâîðÿþùàÿ äëÿ ëþáîãî ϕ ∈ V 1 è ïî÷òè âñåõ t ∈ (0, T ) ðàâåíñòâó

∫Ω

v′ϕdx−n∑

i,j=1

∫Ω

vivj∂ϕj∂xi

dx+ κ∫Ω

∇(v′) : ∇ϕdx+

+ ν

∫Ω

∇v : ∇ϕdx+ κn∑

i,j=1

∫Ω

vi∆vj∂ϕj∂xi

dx = 〈f, ϕ〉

è íà÷àëüíîìó óñëîâèþ v|t=0 = a.Òåîðåìà. Ñóùåñòâóåò õîòÿ áû îäíî ñëàáîå ðåøåíèå íà÷àëüíî-êðàå-

âîé çàäà÷è (1)(2).Ðàáîòà âûïîëíåíà ïðè ôèíàíñîâîé ïîääåðæêå Ìèíèñòåðñòâà îáðàçîâàíèÿ

è íàóêè ÐÔ (ïðîåêò 14.Z50.31.0037).

99


ÏÐÈÌÅÍÅÍÈÅ ÑÒÀÖÈÎÍÀÐÍÎÃÎ ÌÅÒÎÄÀÃÀËÅÐÊÈÍÀ Ê ÎÄÍÎÉ ÊÐÀÅÂÎÉ ÇÀÄÀ×ÅÄËß ÓÐÀÂÍÅÍÈß ÂÛÑÎÊÎÃÎ ÏÎÐßÄÊÀ ÑÌÅÍßÞÙÈÌÑß ÍÀÏÐÀÂËÅÍÈÅÌ ÂÐÅÌÅÍÈ

Ôåäîðîâ Â. Å.

Íàó÷íî-èññëåäîâàòåëüñêèé èíñòèòóò ìàòåìàòèêèÑåâåðî-Âîñòî÷íîãî ôåäåðàëüíîãî óíèâåðñèòåòà èì. Ì.Ê. Àììîñîâà,

ßêóòñê, Ðîññèÿ; [email protected]

Â öèëèíäðè÷åñêîé îáëàñòè ðàññìàòðèâàåòñÿ óðàâíåíèå ïîðÿäêà2s+1 ïî âðåìåíè è âòîðîãî ïîðÿäêà ïî ïðîñòðàíñòâåííûì ïåðåìåííûì.Ñòàðøèé êîýôôèöèåíò óðàâíåíèÿ ìîæåò ìåíÿòü ñâîé çíàê âíóòðè îá-ëàñòè ïðîèçâîëüíûì îáðàçîì. Ïîýòîìó ýòîò êëàññ óðàâíåíèé âêëþ÷àåòâ ñåáÿ ýëëèïòèêî-ïàðàáîëè÷åñêèå óðàâíåíèÿ, óðàâíåíèÿ ñ ìåíÿþùèì-ñÿ íàïðàâëåíèåì âðåìåíè è äðóãèå óðàâíåíèÿ. Äëÿ îäíîãî èç ñëó÷àåâçíàêîîïðåäåëåííîñòè ñòàðøåãî êîýôôèöèåíòà óðàâíåíèÿ íà îñíîâàíèÿõöèëèíäðà ñ ïîìîùüþ ñòàöèîíàðíîãî ìåòîäà Ãàëåðêèíà äîêàçàíà îäíî-çíà÷íàÿ ðåãóëÿðíàÿ ðàçðåøèìîñòü â ïðîñòðàíñòâå Ñîáîëåâà îäíîé êðàå-âîé çàäà÷è äëÿ äàííîãî óðàâíåíèÿ, êîòîðàÿ â ðàáîòå [1] áûëà èññëåäîâà-íà äëÿ ýòîãî æå ñëó÷àÿ ñ ïîìîùüþ íåñòàöèîíàðíîãî ìåòîäà Ãàëåðêèíàâ ñî÷åòàíèè ñ ìåòîäîì ðåãóëÿðèçàöèè. Óñòàíîâëåíà îöåíêà ïîãðåøíî-ñòè ïðèáëèæåííûõ ðåøåíèé îòíîñèòåëüíî òî÷íîãî ðåøåíèÿ çàäà÷è ÷å-ðåç ñîáñòâåííûå çíà÷åíèÿ ñàìîñîïðÿæåííîé ñïåêòðàëüíîé çàäà÷è äëÿêâàçèýëëèïòè÷åñêîãî óðàâíåíèÿ ïîðÿäêà 2s + 2 ïî âðåìåíè. Ñîáñòâåí-íûå ôóíêöèè ýòîé çàäà÷è âûáèðàþòñÿ â êà÷åñòâå áàçèñà ïðè ïîñòðîåíèèïðèáëèæåííûõ ðåøåíèé.

Äëÿ äðóãîãî ñëó÷àÿ çíàêîîïðåäåëåííîñòè ñòàðøåãî êîýôôèöèåíòàóðàâíåíèÿ íà îñíîâàíèÿõ öèëèíäðà òåì æå ñòàöèîíàðíûì ìåòîäîì Ãà-ëåðêèíà âïåðâûå äîêàçàíà îäíîçíà÷íàÿ ðàçðåøèìîñòü ðàññìàòðèâàåìîéçàäà÷è â âåñîâîì ïðîñòðàíñòâå Ñîáîëåâà. Çäåñü â êà÷åñòâå áàçèñà ïðèïîñòðîåíèè ïðèáëèæåííûõ ðåøåíèé âûáèðàþòñÿ ñîáñòâåííûå ôóíêöèèñàìîñîïðÿæåííîé ñïåêòðàëüíîé çàäà÷è äëÿ êâàçèýëëèïòè÷åñêîãî óðàâ-íåíèÿ ïîðÿäêà 2s ïî âðåìåíè. Ïîëó÷åíà îöåíêà ïîãðåøíîñòè ñòàöèî-íàðíîãî ìåòîäà Ãàëåðêèíà è â ýòîì ñëó÷àå.

Çàìå÷àíèå. Îòìåòèì, ÷òî â ðàáîòå [2] ïîëó÷åíû ðåçóëüòàòû, àíà-ëîãè÷íûå âòîðîìó ñëó÷àþ, äëÿ ïåðâîé êðàåâîé çàäà÷è.

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Ðàáîòà âûïîëíåíà ïðè ïîääåðæêå Ìèíèñòåðñòâà îáðàçîâàíèÿ è íàóêè Ðîñ-

ñèéñêîé Ôåäåðàöèè â ðàìêàõ áàçîâîé ÷àñòè ãîñóäàðñòâåííîãî çàäàíèÿ íà âû-

ïîëíåíèå ÍÈÐ íà 20172019 ãã. (ïðîåêò 6069).


1. Egorov I. E. On one boundary value problem for an equation with varying timedirection // Ìàò. çàìåòêè ßÃÓ. 1998. Ò. 5, âûï. 2. Ñ. 7784.

2. Åôèìîâà Å.Ñ. Ïðèìåíåíèå ñòàöèîíàðíîãî ìåòîäà Ãàëåðêèíà ê íåêëàññè-÷åñêîìó óðàâíåíèþ âûñîêîãî ïîðÿäêà ñ ìåíÿþùèìñÿ íàïðàâëåíèåì âðå-ìåíè // Ìàò. çàìåòêè ßÃÓ. 2012. Ò. 19, âûï. 2. Ñ. 3238.

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ÐÅØÅÍÈß ÏÀÐÀÁÎËÈ×ÅÑÊÎÉ ÇÀÄÀ×ÈÑ ÏÐÅÎÁÐÀÇÎÂÀÍÈÅÌ ÏÎÂÎÐÎÒÀ

ÍÀ ÎÊÐÓÆÍÎÑÒÈ

Õàçîâà Þ.À.

Êðûìñêèé ôåäåðàëüíûé óíèâåðñèòåò èì. Â.È. Âåðíàäñêîãî,Òàâðè÷åñêàÿ àêàäåìèÿ, Ñèìôåðîïîëü, Ðîññèÿ;

[email protected]

Èññëåäóþòñÿ ñëó÷àè ðîæäåíèÿ ðåøåíèé òèïà áåãóùåé âîëíû â ïà-ðàáîëè÷åñêîé çàäà÷å íà îêðóæíîñòè ñ ïðåîáðàçîâàíèåì ïîâîðîòà ïðî-ñòðàíñòâåííîé ïåðåìåííîé. Íàéäåíû ïðåäñòàâëåíèÿ äëÿ îðáèòàëüíîóñòîé÷èâîé è íåóñòîé÷èâîé áåãóùèõ âîëí. Äëÿ ïîñòðîåíèÿ ïåðèîäè÷å-ñêèõ ðåøåíèé ïðèìåíÿåòñÿ ìåòîä Ãàëåðêèíà.

Ðàññìàòðèâàåòñÿ ïàðàáîëè÷åñêîå óðàâíåíèå íà îêðóæíîñòè S1:

u = µ4u− u− ΛQu+Λ

6Qu3, u(ϕ+ 2π, t) = u(ϕ, t).

Áóäåì èñïîëüçîâàòü ãàëåðêèíñêóþ àïïðîêñèìàöèþ â âèäå:u =

∑Nk=1(zke

ikϕ + zke−ikϕ). Ñâåäåì èññëåäîâàíèå èñõîäíîé çàäà÷è ê

èññëåäîâàíèþ ñèñòåìû óðàâíåíèé

zk = λkzk + gk(z, z),

zk = λkzk + gk(z, z), k = 1, N.

Ïðè óìåíüøåíèè ïàðàìåòðà µ è åãî ïåðåõîäå ÷åðåç êðèòè÷åñêîå çíà÷å-íèå µ∗1 íóëåâîå ðåøåíèå òåðÿåò óñòîé÷èâîñòü êîëåáàòåëüíûì îáðàçîì. Âðåçóëüòàòå îò íóëÿ îòâåòâëÿåòñÿ ïåðèîäè÷åñêîå ïî t ðåøåíèå òèïà áåãó-ùåé âîëíû. Ïðèìåíÿÿ ìåòîä Ãàëåðêèíà, áûëè ïîñòðîåíû ïðåäñòàâëåíèÿäëÿ ðîæäàþùèõñÿ ïåðèîäè÷åñêèõ ðåøåíèé è èññëåäîâàíà èõ óñòîé÷è-âîñòü.


1. Õàçîâà Þ.À. Ñòàöèîíàðíûå ñòðóêòóðû â ïàðàáîëè÷åñêîé çàäà÷å ñ îòðà-æåíèåì ïðîñòðàíñòâåííîé ïåðåìåííîé // Òàâðè÷åñêèé âåñòíèê èíôîðìà-òèêè è ìàòåìàòèêè. 2015. 3. Ñ. 8295.

2. Õàçîâà Þ.À. Ñòàöèîíàðíûå ñòðóêòóðû â ïàðàáîëè÷åñêîé çàäà÷å ñ îòðà-æåíèåì ïðîñòðàíñòâåííîé ïåðåìåííîé // Àêòóàëüíûå íàïðàâëåíèÿ íàó÷-íûõ èññëåäîâàíèé XXI âåêà: òåîðèÿ è ïðàêòèêà. 2015. Ò. 8-4, 3. Ñ. 314317.

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ÐÅØÅÍÈß ÇÀÄÀ× ÊÎØÈ Â ÃÅÎÌÅÕÀÍÈÊÅ

×àíûøåâ À.È.1,2, Àáäóëèí È.Ì.1, Áåëîóñîâà Î. Å.1

1Èíñòèòóò ãîðíîãî äåëà èì. Í.À. ×èíàêàëà ÑÎ ÐÀÍ,Íîâîñèáèðñê, Ðîññèÿ;

2Íîâîñèáèðñêèé ãîñóäàðñòâåííûé óíèâåðñèòåò ýêîíîìèêè èóïðàâëåíèÿ, Íîâîñèáèðñê, Ðîññèÿ; [email protected]

Ñòðîÿòñÿ àíàëèòè÷åñêèå è ÷èñëåííûå ðåøåíèÿ çàäà÷è Êîøè äëÿ îñ-íîâíûõ óðàâíåíèé ìàòåìàòè÷åñêîé ôèçèêè, âêëþ÷àÿ óïðóãîñòü, ïëà-ñòè÷íîñòü. Âîññòàíàâëèâàþòñÿ íàïðÿæåííî-äåôîðìèðîâàííûå ñîñòîÿ-íèÿ, òåïëîâîå è äð., îïðåäåëÿþòñÿ âíóòðåííÿÿ ñòðóêòóðà òåëà, ñîñðå-äîòî÷åííûå èñòî÷íèêè â íåì.

Ïðèìåðû àíàëèòè÷åñêèõ ðåøåíèé.

1. Óðàâíåíèå Ëàïëàñà äëÿ ïîëóïëîñêîñòè: ∂2u∂x2 + ∂2u

∂y2 = 0.

Ãðàíè÷íûå óñëîâèÿ: u|y=0 = 2g1(x), ∂u∂y

∣∣∣y=0

= 2g′2(x).

Ðåøåíèå u = f(z)+f(z), ãäå f(z) = g1(z)− ig2(z), z = x+ iy, g1, g2 çàäàííûå ãðàíè÷íûå ôóíêöèè.

2. Âîëíîâîå óðàâíåíèå äëÿ ïîëóáåñêîíå÷íîãî ñòåðæíÿ: ∂2u∂x2 = 1

a2∂2u∂t2 .

Ãðàíè÷íûå óñëîâèÿ u|x=0 = 2β(t), ∂u∂x

∣∣x=0

= 2γ′(t)a .

Ðåøåíèå u = [β(t− x/a) + β(t+ x/a)] + [γ(t+ x/a)− γ(t− x/a)] , ãäåβ, γ çàäàííûå ãðàíè÷íûå ôóíêöèè.

3. Ïëîñêàÿ çàäà÷à òåîðèè óïðóãîñòè äëÿ ïîëóïëîñêîñòè.Ãðàíè÷íûå óñëîâèÿ ïðè y = 0: σy = f1(x), τxy = f2(x), ux = f3(x),

uy = f4(x), ãäå f1, f2, f3, f4 ïðîèçâîëüíûå ôóíêöèè.Ðåøåíèå äëÿ ïîòåíöèàëîâ Êîëîñîâà Ìóñõåëèøâèëè:

ϕ(z) =

∫[f1(z)− if2(z)]dz

1 + ℵ+

2µ

1 + ℵ[f3(z) + if4(z)] + C1,

ψ(z) =ℵ

1 + ℵ

∫[f1(z) + if2(z)]dz − z

1 + ℵ[f1(z)− if2(z)]

− 2µ

1 + ℵ[f3(z)− if4(z) + z(f ′3(z) + if ′4(z))] + C2, ℵ = 3− 4ν

(C1, C2 ïîñòîÿííûå). Èññëåäóåòñÿ óñòîé÷èâîñòü ðåøåíèé îò âõîäíûõäàííûõ.

103


ÓÐÀÂÍÅÍÈÅ ÊäÔ ÏßÒÎÃÎ ÏÎÐßÄÊÀ

×óåøåâà Í.À.

Êåìåðîâñêèé ãîñóäàðñòâåííûé óíèâåðñèòåò, Êåìåðîâî, Ðîññèÿ;[email protected]

Â ñòàòüå Z. Bing-Yu, Z. Deqin [1] ðàññìàòðèâàåòñÿ íà÷àëüíàÿ çàäà÷àäëÿ óðàâíåíèÿ ÊäÔ ïÿòîãî ïîðÿäêà

ut − uxxxxx = c1uux + c2u2ux + 2buxuxx + buuxxx. (1)

Â ýòîé çàìåòêå ðàññìîòðèì çàäà÷ó, ñôîðìóëèðóåì òåîðåìó è ïðèâå-ä¼ì íåñêîëüêî ïðèìåðîâ ðåøåíèé óðàâíåíèÿ (1).

Çàäà÷à. Ïóñòü äàí ïðÿìîóãîëüíèê Π = (x, t) ∈ R2 : x ∈ (0, a), t ∈(0, b), â êîòîðîì çàäàíî óðàâíåíèå (1), Γ ãðàíèöà ïðÿìîóãîëüíèêà,n = (nx, nt) âåêòîð âíóòðåííåé íîðìàëè ê ãëàäêèì ó÷àñòêàì ãðàíèöûΓ. Ïóñòü ðåøåíèå óðàâíåíèÿ (1) óäîâëåòâîðÿåò ãðàíè÷íûì óñëîâèÿì

u|x=0,x=a = ux|x=0,x=a = uxx|x=0 = u|t=0 = 0. (2)

Òåîðåìà. Ïóñòü êîýôôèöèåíòû äèôôåðåíöèàëüíîãî óðàâíåíèÿ (1)c1, c2, b áóäóò êîìïëåêñíûìè èëè âåùåñòâåííûìè ïîñòîÿííûìè. Òîãäà

ðåøåíèå ãðàíè÷íîé çàäà÷è (2) äëÿ óðàâíåíèÿ (1) ïî÷òè âñþäó â ïðÿìî-óãîëüíèêå Π áóäåò ðàâíî íóëþ.

Ïðèìåð. Åñëè: 1) c1 = 12b, c2 = 0, b 6= 0, òî ðåøåíèåì óðàâ-íåíèÿ (1) áóäåò àíàëèòè÷åñêàÿ íà êîìïëåêñíîé ïëîñêîñòè C ôóíêöèÿu(z) = − 4b+16−i

8b + sin2 (x+ it), ãäå z = x+ it;2) c1 = 12b, c2 = 0, b 6= 0, òî ðåøåíèåì óðàâíåíèÿ (1) áóäåò ôóíêöèÿu(z) = − 4b+16+i

8b + sin2 (x− it) (z = x− it), êîòîðàÿ R-äèôôåðåíöèðóåìàíà R2, íî íå C-äèôôåðåíöèðóåìà íà âñåé êîìïëåêñíîé ïëîñêîñòè C;3) c1 = 0, c2 = 0, b = −1/42, òî ðåøåíèåì óðàâíåíèÿ (1) áóäåò ðàçðûâ-

íàÿ íà îñè x âåùåñòâåííàÿ ôóíêöèÿ u(x, t) = x3

t ∈ C∞ (R2 \ t = 0

);

4) c1 = 0, c2 = 120, b = −20, òî ðåøåíèåì óðàâíåíèÿ (1) áóäåò àíàëèòè-÷åñêàÿ íà êîìïëåêñíîé ïëîñêîñòè C áåç òî÷åê zk = π

2 +πk, k ∈ Z, ôóíê-öèÿ u(z) = a+tan2 (x+ it), a = 2

3 + 160

4√

7300 ·eiϕ, ϕ = 12

(π − 1

2 arctan 38

).

Òî÷êè zk ÿâëÿþòñÿ ïîëþñàìè âòîðîãî ïîðÿäêà ýòîãî ðåøåíèÿ óðàâíå-íèÿ (1).


1. Bing-Yu Z., Deqin Z. Initial boundary value problem of the Hamiltonian fth-order KdV equation on a bounded domain // Adv. Dier. Equ. 2016. V. 21,No 9/10. P. 9771000.

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ÈÑÑËÅÄÎÂÀÍÈÅ ÐÀÇÐÅØÈÌÎÑÒÈÊÐÀÅÂÛÕ ÇÀÄÀ× Ñ ÄÎÏÎËÍÈÒÅËÜÍÛÌÈ

ÓÑËÎÂÈßÌÈ ÑÎÏÐßÆÅÍÈß

Øàäðèíà Í.Í.

Áóðÿòñêèé ãîñóäàðñòâåííûé óíèâåðñèòåò, Óëàí-Óäý, Ðîññèÿ;[email protected]

Â ðàáîòå èññëåäóåòñÿ âëèÿíèå ïàðàìåòðîâ, âõîäÿùèõ â óñëîâèÿ ñî-ïðÿæåíèÿ, à òàêæå ïàðàìåòðîâ, îïðåäåëÿþùèõ ãðàíè÷íûå óñëîâèÿ, íàåäèíñòâåííîñòü è íååäèíñòâåííîñòü, ñóùåñòâîâàíèå è íåñóùåñòâîâàíèåðåøåíèé íåêîòîðîé çàäà÷è ñîïðÿæåíèÿ äëÿ óðàâíåíèÿ ýëëèïòè÷åñêîãîòèïà.

Ðàññìàòðèâàåòñÿ ñëåäóþùàÿ çàäà÷à: íàéòè ôóíêöèþ u(x, y), ÿâëÿþ-ùóþñÿ â öèëèíäðàõ Q1 è Q2 ðåøåíèåì óðàâíåíèÿ

∆u = λu,

è òàêóþ, ÷òî äëÿ íåå âûïîëíÿþòñÿ óñëîâèÿ

u(x, y)|S1∪S2= 0,

u(x, a) = u(x, 1) = 0, x ∈ Ω,

à òàêæå óñëîâèÿ

u(x,−0) = βu(x,+0), uy(x,+0) = αuy(x,−0)

(α, β çàäàííûå äåéñòâèòåëüíûå ÷èñëà).Ïîêàçàíî, ÷òî ñâîéñòâî äàííîãî äåéñòâèòåëüíîãî ÷èñëà λ0 áûòü èëè

íå áûòü ñîáñòâåííûì ÷èñëîì äàííîé çàäà÷è, ñâîéñòâî îòðèöàòåëüíîãî÷èñëà a áûòü èëè íå áûòü êðèòè÷åñêèì äëÿ òîé æå çàäà÷è îïðåäåëÿþòñÿïðîèçâåäåíèåì ïàðàìåòðîâ α è β.


1. Øàäðèíà Í.Í. Î âëèÿíèè ïàðàìåòðîâ íà ðàçðåøèìîñòü íåêîòîðûõ çà-äà÷ ñîïðÿæåíèÿ äëÿ ýëëèïòè÷åñêèõ óðàâíåíèé // Ñèá. ýëåêòðîí. ìàò. èçâ.2016. Ò. 13. Ñ. 411425.

105


ËÎÊÀËÜÍÀß ÀÑÈÌÏÒÎÒÈ×ÅÑÊÀßÝÊÂÈÂÀËÅÍÒÍÎÑÒÜ È ÅÅ ÏÐÈÌÅÍÅÍÈÅÊ ÈÑÑËÅÄÎÂÀÍÈÞ ÓÑÒÎÉ×ÈÂÎÑÒÈ

ÏÎ ×ÀÑÒÈ ÏÅÐÅÌÅÍÍÛÕ

Øàìàíàåâ Ï.À., ßçîâöåâà Î.Ñ.

Ìîðäîâñêèé ãîñóäàðñòâåííûé óíèâåðñèòåò èì. Í.Ï. Îãàð¼âà,Ñàðàíñê, Ðîññèÿ; [email protected]

Èäåÿ ðàçáèåíèÿ ìíîæåñòâà ñèñòåì îáûêíîâåííûõ äèôôåðåíöèàëü-íûõ óðàâíåíèé íà êëàññû ýêâèâàëåíòíîñòè íà îñíîâå àñèìïòîòè÷åñêîãîïîâåäåíèÿ èõ ðåøåíèé ïðèíàäëåæèò À.Ì. Ëÿïóíîâó. Â ðàáîòàõ [13]äëÿ êëàññèôèêàöèè íåëèíåéíûõ ñèñòåì ââåäåíû ïîíÿòèÿ ïîêîìïîíåíò-íîé àñèìïòîòè÷åñêîé ýêâèâàëåíòíîñòè ïî Áðàóýðó è Ëåâèíñîíó îòíî-ñèòåëüíî íåêîòîðûõ ýòàëîííûõ ôóíêöèé. Ââåäåííûå îïðåäåëåíèÿ ëî-êàëüíîé ïîêîìïîíåíòíîé àñèìïòîòè÷åñêîé ýêâèâàëåíòíîñòè ïîçâîëÿþòðàñïðîñòðàíèòü ìåòîäèêó íà áîëåå øèðîêèé êëàññ íåëèíåéíûõ ñèñòåì,÷åì â ðàáîòàõ [13], ïîëó÷åíû äîñòàòî÷íûå óñëîâèÿ ýêâèâàëåíòíîñòèäëÿ íåëèíåéíûõ ñèñòåì îáûêíîâåííûõ äèôôåðåíöèàëüíûõ óðàâíåíèéñ âîçìóùåíèÿìè â âèäå âåêòîðíûõ ïîëèíîìîâ.

Â íàñòîÿùåé ðàáîòå ïðîâåäåíî èññëåäîâàíèå óñòîé÷èâîñòè ïî ÷à-ñòè ïåðåìåííûõ íåòðèâèàëüíîãî ïîëîæåíèÿ ðàâíîâåñèÿ ñèñòåìû íåëè-íåéíûõ óðàâíåíèé, ïðåäñòàâëÿþùèõ ñîáîé êèíåòè÷åñêóþ ìîäåëü õèìè-÷åñêîé ðåàêöèè, ñâåäåííîå ê èññëåäîâàíèþ óñòîé÷èâîñòè òðèâèàëüíî-ãî ïîëîæåíèÿ ðàâíîâåñèÿ ëèíåéíîãî ïðèáëèæåíèÿ ñèñòåìû íà îñíîâà-íèè óñòàíîâëåíèÿ ëîêàëüíîé ïîêîìïîíåíòíîé àñèìïòîòè÷åñêîé ýêâèâà-ëåíòíîñòè ïî Áðàóýðó. Íåëèíåéíàÿ ñèñòåìà ïðåäñòàâëÿåò ñîáîé ñèñòåìóíåëèíåéíûõ äèôôåðåíöèàëüíûõ óðàâíåíèé ñ âîçìóùåíèÿìè â âèäå âåê-òîðíûõ ïîëèíîìîâ.

Ìåòîä èññëåäîâàíèÿ îñíîâàí íà ïîñòðîåíèè îïåðàòîðà, ïåðåâîäÿùå-ãî ðåøåíèÿ íåëèíåéíîé ñèñòåìû â ðåøåíèÿ åå ëèíåéíîãî ïðèáëèæåíèÿ.Â ñîîòâåòñòâèè ñ ìåòîäèêîé, ïðåäñòàâëåííîé â [1], äîêàçàòåëüñòâî ñó-ùåñòâîâàíèÿ îïåðàòîðà îñíîâàíî íà ïðèíöèïå Øàóäåðà î íåïîäâèæ-íîé òî÷êå. Äîêàçàòåëüñòâî ïðîâîäèòñÿ ñ èñïîëüçîâàíèåì îöåíîê ýëåìåí-òîâ ôóíäàìåíòàëüíîé ìàòðèöû ëèíåéíîãî ïðèáëèæåíèÿ è âîçìóùåíèéíåëèíåéíîé ñèñòåìû [2].

Ñóùåñòâîâàíèå îïåðàòîðà ïîçâîëÿåò ïîñòðîèòü îòîáðàæåíèå, ñâÿçû-âàþùåå íà÷àëüíûå òî÷êè èññëåäóåìîé ñèñòåìû è åå ëèíåéíîãî ïðèáëè-

106


æåíèÿ, ÷òî îáåñïå÷èâàåò ëîêàëüíóþ ïîêîìïîíåíòíóþ àñèìïòîòè÷åñêóþýêâèâàëåíòíîñòü ïî Áðàóýðó.


1. Âîñêðåñåíñêèé Å.Â. Àñèìïòîòè÷åñêèå ìåòîäû: òåîðèÿ è ïðèëîæåíèÿ. Ñà-ðàíñê: ÑÂÌÎ, 2000.

2. Âîñêðåñåíñêèé Å.Â., Àðòåìüåâà Å.Í., Áåëîãëàçîâ Â.À., Ìóðþìèí Ñ.Ì.Êà÷åñòâåííûå è àñèìïòîòè÷åñêèå ìåòîäû èíòåãðèðîâàíèÿ äèôôåðåíöè-àëüíûõ óðàâíåíèé. Ñàðàíñê: Èçä-âî Ñàðàòîâñê. óí-òà (Ñàðàíñêèé ôèëè-àë), 1988.

3. Âîñêðåñåíñêèé Å.Â. Ìåòîäû ñðàâíåíèÿ â íåëèíåéíîì àíàëèçå. Ñàðàíñê:Èçä-âî Ñàðàòîâñê. óí-òà (Ñàðàíñêèé ôèëèàë), 1990.

4. Áûëîâ Á.Ô., Âèíîãðàä Ð.Ý., Ãðîáìàí Ä.Ì., Íåìûöêèé Â.Â. Òåîðèÿ ïîêà-çàòåëåé Ëÿïóíîâà è åå ïðèëîæåíèÿ ê âîïðîñàì óñòîé÷èâîñòè. Ì.: Íàóêà,1966.

107


ÂÍÅÖÅÍÒÐÅÍÍÎÅ ÍÀÃÐÓÆÅÍÈÅ ØÒÀÌÏÀÑ ÄÂÓÑÂßÇÍÎÉ ÎÁËÀÑÒÜÞ ÊÎÍÒÀÊÒÀ

Øèøêàíîâà À.À.

Çàïîðîæñêèé íàöèîíàëüíûé òåõíè÷åñêèé óíèâåðñèòåò,Çàïîðîæüå, Óêðàèíà; [email protected]

Ìàòåìàòè÷åñêàÿ ìîäåëü çàäà÷è îöåíêè íàïðÿæåííî-äåôîðìèðóåìî-ãî ñîñòîÿíèÿ è ïðîãíîçèðîâàíèÿ îòðûâà øòàìïà (ñòîéêè èëè îïîðûèíæåíåðíûõ êîíñòðóêöèé) ïðè âíåöåíòðåííîì íàãðóæåíèè âêëþ÷àåòíåîáõîäèìîñòü ðåøåíèÿ ñèñòåìû óðàâíåíèé, ñîñòîÿùåé èç äâóìåðíîãîèíòåãðàëüíîãî óðàâíåíèÿ òèïà Ôðåäãîëüìà ñî ñëàáîé îñîáåííîñòüþ, âû-ðàæàþùåãî çàâèñèìîñòü ìåæäó âåðòèêàëüíûìè ïåðåìåùåíèÿìè è íîð-ìàëüíûìè íàïðÿæåíèÿìè, è åùå òðåõ óðàâíåíèé ðàâíîâåñèÿ.

Ðàçðàáîòàíî ÷èñëåííî-àíàëèòè÷åñêîå ðåøåíèå, èñïîëüçóþùåå ðåãó-ëÿðèçàöèþ îñíîâíîãî óðàâíåíèÿ. Ïàðàìåòð ðåãóëÿðèçàöèè èìååò îïðå-äåëåííûé ìåõàíè÷åñêèé ñìûñë êîýôôèöèåíòà øåðîõîâàòîñòè óïðóãî-ãî ïîëóïðîñòðàíñòâà [1]. Çàäà÷à î âäàâëèâàíèè íåêðóãîâûõ êîëüöåâûõøòàìïîâ ñ èñïîëüçîâàíèåì ðàçëîæåíèÿ ïîòåíöèàëîâ ïðîñòîãî ñëîÿ ñâå-äåíà ê ðåøåíèþ ïîñëåäîâàòåëüíîñòè àíàëîãè÷íûõ êîíòàêòíûõ çàäà÷âäàâëèâàíèÿ øòàìïîâ â ôîðìå êðóãîâîãî êîëüöà, ïîñëå ÷åãî ïðèìåíÿ-þòñÿ êâàäðàòóðíûå ôîðìóëû. Äëÿ ìíîãîñâÿçíîé îáëàñòè ïðè ó÷åòå øå-ðîõîâàòîñòè ïðåäëàãàåòñÿ ÷èñëåííîå ðåøåíèå, èñïîëüçóþùåå êóáàòóð-íûå ôîðìóëû äëÿ äèñêðåòèçàöèè äâóìåðíîãî èíòåãðàëüíîãî îïåðàòîðà,â ðåçóëüòàòå ÷åãî ïîëó÷åíà ñèñòåìà àëãåáðàè÷åñêèõ óðàâíåíèé, äëÿ ðå-øåíèÿ êîòîðîé ïðèìåíÿåì ìåòîä ïîñëåäîâàòåëüíûõ ïðèáëèæåíèé.

Èññëåäîâàí êîíêðåòíûé ñëó÷àé âíåöåíòðåííîãî íàãðóæåíèÿ øòàì-ïà ñ äâóñâÿçíûì îñíîâàíèåì, îãðàíè÷åííûì ãëàäêèìè êðèâûìè, áëèç-êèìè ê êâàäðàòó. Îïðåäåëåíà çîíà óñòîé÷èâîñòè. Ïðè âûõîäå ëèíèèäåéñòâèÿ ñèëû çà ïðåäåëû ýòîé çîíû ïîäîøâà øòàìïà çàéìåò íàêëîí-íîå ïîëîæåíèå è ïðîèçîéäåò îòðûâ îò ãðóíòà, ÷òî ìîæåò ïðèâåñòè êàâàðèè. Íàéäåíû âåëè÷èíà çàãëóáëåíèÿ, óãîë íàêëîíà è ôóíêöèÿ ðàñ-ïðåäåëåíèÿ íîðìàëüíûõ äàâëåíèé ïîä øòàìïîì.


1. Shyshkanova G., Zaytseva T., Frydman O. The analysis of manufacturingerrors eect on contact stresses distribution under the ring parts deformedasymmetrically // Metallurgical and Mining Industry. 2015. No 7. P. 352357.

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ÎÁ ÎÄÍÎÉ ÂÀÐÈÀÖÈÎÍÍÎÉÇÀÄÀ×Å ÌÅÕÀÍÈÊÈ

ÊÎÌÏÎÇÈÖÈÎÍÍÛÕ ÌÀÒÅÐÈÀËÎÂ

Ùåðáàêîâ Â.Â.

Èíñòèòóò ãèäðîäèíàìèêè èì. Ì.À. Ëàâðåíòüåâà ÑÎ ÐÀÍ,Íîâîñèáèðñêèé ãîñóäàðñòâåííûé óíèâåðñèòåò, Íîâîñèáèðñê, Ðîññèÿ;

[email protected]

Ðàññìàòðèâàåòñÿ âàðèàöèîííàÿ çàäà÷à î ðàâíîâåñèè êîìïîçèöèîííî-ãî ìàòåðèàëà, ñîñòîÿùåãî èç óïðóãîé ìàòðèöû è òîíêîãî óïðóãîãî âî-ëîêíà, íàïðÿæåííî-äåôîðìèðîâàííîå ñîñòîÿíèå êîòîðîãî ìîäåëèðóåò-ñÿ óðàâíåíèÿìè Òèìîøåíêî [1]. Ïðåäïîëàãàåòñÿ, ÷òî âîëîêíî ÷àñòè÷íîîòñëàèâàåòñÿ îò ìàòðèöû, ÷òî ïðèâîäèò ê ñóùåñòâîâàíèþ ìåæôàçíîéòðåùèíû. Íà áåðåãàõ òðåùèíû çàäàíû óñëîâèÿ îäíîñòîðîííåãî îãðà-íè÷åíèÿ, ïðåïÿòñòâóþùèå âçàèìíîìó ïðîíèêàíèþ áåðåãîâ. Èçó÷àþòñÿâîçíèêàþùèå â ìåõàíèêå ðàçðóøåíèÿ âîïðîñû, ñâÿçàííûå ñ àíàëèçîìñèíãóëÿðíîñòè ïîëÿ íàïðÿæåíèé â âåðøèíå òðåùèíû. Â ÷àñòíîñòè, âû-ïèñàíà ôîðìóëà äëÿ ñêîðîñòè âûñâîáîæäåíèÿ ýíåðãèè ïðè ïðîäâèæå-íèè òðåùèíû âäîëü íàïðàâëåíèÿ àðìèðîâàíèÿ è ïîñòðîåíû èíòåãðàëûýíåðãèè, èíâàðèàíòíûå îòíîñèòåëüíî çàìêíóòîãî êîíòóðà, îõâàòûâàþ-ùåãî îäíó èëè îáå âåðøèíû òðåùèíû [2].


1. Itou H., Khludnev A.M. On delaminated thin Timoshenko inclusions insideelastic bodies // Math. Methods Appl. Sci. 2016. V. 39, No 17. P. 49804993.

2. Shcherbakov V.V. The Grith formula and J-integral for elastic bodies withTimoshenko inclusions // Z. Angew. Math. Mech. 2016. V. 96, No 11. P. 13061317.

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ÝÊÑÏÎÍÅÍÖÈÀËÜÍÀß ÓÑÒÎÉ×ÈÂÎÑÒÜÄÈÔÔÅÐÅÍÖÈÀËÜÍÛÕ ÓÐÀÂÍÅÍÈÉÑ ÇÀÏÀÇÄÛÂÀÞÙÈÌ ÀÐÃÓÌÅÍÒÎÌ

Ûñêàê Ò.Ê.


Â ðàáîòå ðàññìàòðèâàåòñÿ ñèñòåìà äèôôåðåíöèàëüíûõ óðàâíåíèé ñðàñïðåäåëåííûì çàïàçäûâàíèåì ñ ïàðàìåòðîì

d

dty(t) = µA(t)y(t) +

t∫t−τ

B(t, t− s)y(s)ds, t > 0,

ãäå µ > 0 ïàðàìåòð, τ > 0 çàïàçäûâàíèå, A(t) íåïðåðûâíàÿT -ïåðèîäè÷åñêàÿ ìàòðèöà, B(t, ξ) íåïðåðûâíàÿ ïî ñîâîêóïíîñòè ïå-ðåìåííûõ ìàòðèöà, T -ïåðèîäè÷åñêàÿ ïî ïåðåìåííîé t. Ïðåäïîëàãàåò-ñÿ, ÷òî ñïåêòð ìàòðèöû A(t) ëåæèò â ëåâîé ïîëóïëîñêîñòè λ ∈ C :Reλ < 0 ïðè âñåõ t ∈ [0, T ]. Â [1] â ñëó÷àå B(t, ξ) ≡ 0 è â [2] â ñëó÷àåóðàâíåíèÿ ñ çàïàçäûâàíèåì ïîêàçàíî, ÷òî íóëåâîå ðåøåíèå ýêñïîíåíöè-àëüíî óñòîé÷èâî ïðè âñåõ äîñòàòî÷íî áîëüøèõ ïàðàìåòðàõ µ. Â äàííîéðàáîòå óêàçàíû óñëîâèÿ íà ïàðàìåòð µ, ïðè êîòîðûõ íóëåâîå ðåøåíèåýêñïîíåíöèàëüíî óñòîé÷èâî, óñòàíîâëåíà îöåíêà, õàðàêòåðèçóþùàÿ ýêñ-ïîíåíöèàëüíóþ ñêîðîñòü óáûâàíèÿ ðåøåíèé íà áåñêîíå÷íîñòè. Ïðè èñ-ñëåäîâàíèè áûëà èñïîëüçîâàíà ìîäèôèêàöèÿ ôóíêöèîíàëà Ëÿïóíîâà Êðàñîâñêîãî, ââåäåííîãî â [3, 4].




1. Äàëåöêèé Þ.Ë., Êðåéí Ì.Ã. Óñòîé÷èâîñòü ðåøåíèé äèôôåðåíöèàëüíûõóðàâíåíèé â áàíàõîâûõ ïðîñòðàíñòâàõ. Ì.: Íàóêà, 1970.

2. Demidenko G.V., Matveeva I. I. Asymptotic stability of solutions to a class oflinear time-delay systems with periodic coecients and a large parameter //J. Inequal. Appl. 2015. V. 2015, No 331. P. 110.

3. Demidenko G.V. Stability of solutions to linear dierential equations of neutraltype // J. Anal. Appl. 2009. V. 7, No 3. P. 119130.

4. Äåìèäåíêî Ã.Â., Ìàòâååâà È.È. Îá îöåíêàõ ðåøåíèé ñèñòåì äèôôåðåí-öèàëüíûõ óðàâíåíèé íåéòðàëüíîãî òèïà ñ ïåðèîäè÷åñêèìè êîýôôèöèåí-òàìè // Ñèá. ìàò. æóðí. 2014. Ò. 55, 5. Ñ. 10591077.

110


ON LONG TIME BEHAVIOR OF SOLUTIONSOF THE NORMALIZED RICCI FLOW ON SPECIAL

GENERALIZED WALLACH SPACES

Abiev N.A.

Taraz State University named after M.Kh. Dulaty,Taraz, Republic of Kazakhstan; [email protected]

The authors of [1] obtained that the normalized Ricci ow equationg(t) = −2Ricg + 2n−1g(t)Sg, where g(t) means a 1-parameter family ofRiemannian metrics, Ricg and Sg are respectively the Ricci tensor andthe scalar curvature of g, can be reduced to the following system of ODEsin the case of some very specic generalized Wallach spaces:

w1 = (w1 − 1)(2w1 −w1w2 −w2), w2 = (w2 − 1)(2w2 −w1w2 −w1). (1)

We will discuss some results of [2] relating to asymptotic propertiesof solutions of (1) with respect to a connected domain in R2 bounded bythe curves

w21w

22 + w2

1 − w22 − 4w2

1w2 = 0,

w21w

22 − w2

1 + w22 − 4w1w

22 = 0,

w21w

22 − w2

1 − w22 + 4w1w2 = 0.

Note that geometrically such a domain describes the set of invariantRiemannian metrics on generalized Wallach spaces under considerationwhich admit positive Ricci curvature.

The author was supported by the Grant of MES of the Republic of Kazakhstan

for 20152017 (project no. 1452/GF4).

REFERENCES

1. Abiev N.A., Nikonorov Yu.G., The evolution of positively curved invariantRiemannian metrics on the Wallach spaces under the Ricci ow, Ann. GlobalAnal. Geom., 50, No. 1, 6584 (2016).

2. Abiev N.A., On evolution of invariant Riemannian metrics on one class ofgeneralized Wallach spaces under the normalized Ricci ow [in Russian],Matematicheskie trudy, 20, No. 1, 320 (2017).

111


A CLASS OF ELECTROMAGNETIC p(x, t)-CURLSYSTEMS: EXISTENCE AND UNIQUENESS,BLOW-UP AND FINITE TIME EXTINCTION

Antontsev S.N.

University of Lisbon, Lisbon, Portugal;Lavrentyev Institute of Hydrodynamics SB RAS, Novosibirsk, Russia;

[email protected]

We consider in QT = Ω×(0, T ), Ω ⊂ R3, the following class of p(x, t)-curlsystems arising in electromagnetism

∂th +∇×(|∇×h|p(x,t)−2∇×h

)= f(h), ∇·h = 0 in QT , (1)

|∇×h|p(x,t)−2∇×h× n = 0, h · n = 0 on ΣT , (2)

h( · , 0) = h0 in Ω, (3)

where h is the unknown magnetic eld and h0 is a given function. Thegiven log-continuous function p(x, t) satises 6

5 < p− ≤ p(x, t) ≤ p+ < ∞.The nonlinear function f(h) can model either a source term of the type

f(h) = h

(∫Ω

|h|2)σ−2

2

with σ > 1 or a sink term f(h) = −h(∫

Ω

|h|2)−λ

with λ > 0. We prove existence and uniqueness of a solution for (1)(3).Blow-up of local solutions is studied in the case of source term and

nite time extinction of the solution is proved in the case of sink term.The detailed proofs can be found in [13] (joint work with F. Miranda andL. Santos).

The author was supported by the Russian Science Foundation (project no. 15-

11-20019).

REFERENCES

1. Antontsev S.N., Shmarev S. I., Evolution PDEs with Nonstandard GrowthConditions: Existence, Uniqueness, Localization, Blow-up; Series: AtlantisStudies in Dierential Equations, Vol. 4, Atlantis Press, Paris, 2015.

2. Antontsev S.N., Miranda F., Santos L., A class of electromagnetic p-curlsystems: blow-up and nite time extinction, Nonlinear Anal., Theory MethodsAppl., 75, 39163929 (2012).

3. Antontsev S.N., Miranda F., Santos L., Blow-up and nite time extinction forp(x, t)-curl systems arising in electromagnetism, J. Math. Anal. Appl., 440,300322 (2016).

112


TOPOLOGICAL BARRIERSFOR LOCALLY HOMEOMORPHIC

QUASIREGULAR MAPPINGS IN 3-SPACE

Apanasov B.N.

University of Oklahoma, Norman, USA; [email protected]

Here we address three sides of the M.A. Lavrentiev problem and theZorich map with an essential singularity at innity: on locally homeomorphicspatial quasiregular mappings dened in the (almost) whole sphere S3,mapping it onto the whole sphere S3, and their essential (topological) singu-larities. We construct a new type of such mappings dened in the 3-sphere S3

except a dense Cantor subset in S2 ⊂ S3 as mappings equivariant with thestandard conformal action of uniform hyperbolic lattices in the unit 3-balland its complement in S3 (by using our non-trivial compact 4-dimensionalcobordisms).

REFERENCES

1. Apanasov B., Nontriviality of Teichmuller space for Kleinian group in space,in: Riemann Surfaces and Related Topics: Proc. 1978 Stony Brook Conference(Ann. Math. Stud., Vol. 97), Princeton Univ. Press, 1981, pp. 2131.

2. Apanasov B., Tetenov A., Nontrivial cobordisms with geometrically nitehyperbolic structures, J. Dier. Geom., 28, 407422 (1988).

3. Apanasov B., Nonstandard uniformized conformal structures on hyperbolicmanifolds, Invent. Math., 105, 137152 (1991).

4. Apanasov B., Conformal Geometry of Discrete Groups and Manifolds(de Gruyter Expo. Math., Vol. 32), W. de Gruyter, Berlin, New York (2000).

5. Apanasov B., Quasisymmetric embeddings of a closed ball inextensible inneighborhoods of any boundary points, Ann. Acad. Sci. Fenn., Ser. AI, 14,243255 (1989).

6. Apanasov B., Group actions, Teichmuuller spaces and cobordisms,Lobachevskii J. Math., 38, 213228 (2017).

7. Apanasov B., Topological barriers for locally homeomorphic quasiregularmappings in 3-space, arxiv.org/abs/1510.08951.

113


ON THE UNIQUE SOLVABILITY OF PERIODICPROBLEM FOR THE SOBOLEV-TYPE PARTIAL

DIFFERENTIAL EQUATION

Assanova A.T.

Institute of Mathematics and Mathematical Modeling MES RK,Almaty, Republic of Kazakhstan; [email protected]

Consider the periodic problem for the Sobolev-type dierential equationof the third order on the domain Ω = [0, T ]× [0, ω]:

∂3u

∂x2∂t= A(t, x)

∂2u

∂x2+B(t, x)

∂2u

∂x∂t+ C(t, x)

∂u

∂x+D(t, x)u+ f(t, x), (1)

∂u(0, x)

∂x=∂u(T, x)

∂x+ ϕ(x), x ∈ [0, ω], (2)

u(t, 0) = ψ1(t), t ∈ [0, T ], (3)

∂u(t, x)

∂x

∣∣∣x=0

=∂u(t, x)

∂x

∣∣∣x=ω

+ψ2(t), t ∈ [0, T ], (4)

where u(t, x) is unknown function, functions A(t, x), B(t, x), C(t, x),D(t, x),and f(t, x) are continuous on Ω, function ϕ(x) is continuously dierentiableon [0, ω], functions ψ1(t) and ψ2(t) are continuously differentiable on [0, T ].

Periodic and nonlocal problems for the Sobolev-type partial dierentialequations appear in various physical processes [1].

In present communication, we investigate conditions of existence anduniqueness of a classical periodic solution to problem (1)(4). For this goalwe use the method of introduction of new functions [2]. Problem (1)(4)is reduced to equivalent problem consisting of periodic problem for hyper-bolic equation with unknown function and integral relation. Periodic prob-lem for hyperbolic equation is studied by method of introduction specialfunctional parameters [3]. This problem reduced to an equivalent probleminvolving Goursat problem for hyperbolic equation with parameters andperiodic problems for ordinary dierential equations. Algorithms for ndingapproximate solutions to equivalent problems are constructed and it isproved their convergence.

Also we propose the approach for nding periodic solution to problem(1)(4) based on these algorithms. Sucient conditions of existence unique

114


periodic solution for the Sobolev-type partial dierential equations (1)(4)are established in terms of coecients of equation (1) and numbers T , ω.

The author was supported by the Grant of Ministry of Education and Sciences

of the Republic of Kazakhstan (project no. 0822/ΓΦ4).

REFERENCES

1. Soltanalizadeh V., Roohani Ghehsareh H., Abbasbandy S., A super accurateshifted Tau method for numerical computation of the Sobolev-type dierentialequation with nonlocal boundary condition, Appl. Math. Comput., 236, 683692 (2014).

2. Asanova A.T., Dzhumabaev D. S., Well-posedness of nonlocal boundary valueproblems with integral condition for the system of hyperbolic equations,J. Math. Anal. Appl., 402, No. 1, 167178 (2013).

3. Assanova A.T., Periodic solutions in the plane of systems of second-orderhyperbolic equations, Math. Notes, 101, No. 1, 1726 (2017).

115


CONFORMAL INVARIANCEOF THE ELASTOSTATICS EQUATIONS

Chirkunov Yu.A.

Novosibirsk State University of Architecture and Civil Engineering,Novosibirsk, Russia; [email protected]

We fullled a group foliation of the system of n-dimensional (n > 1)Lame equations of the classical static theory of the elasticity with respectto the innite subgroup contained in normal subgroup of main group of thissystem. It permitted us to move from the Lame equations to the equiva-lent unication of two rst-order systems: automorphic and resolving. Weobtained a general solution of the automorphic system. This solution isn-dimensional analogue of the KolosovMuskhelishvili formula. We foundthe main Lie group of transformations of the resolving system of this groupfoliation. It turned out that in two-dimensional and three-dimensional cases,which have a physical meaning, this system is conformally invariant, whilethe Lame equations admit only a group of the similarities of Euclideanspace. This is a big success, since in the method of group foliation resolvingequations usually inherit Lie symmetries subgroup of the full symmetrygroup that was not used for the foliation. In three-dimensional case for thesolutions of the resolving system we found the general form of the trans-formations similar to the Kelvin transformation, that are the consequenceof the conformal invariance of the resolving system. In three-dimensionalcase with the help of the complex dependent and independent variables, theresolving system is written as a simple complex system. For this complexsystem, all the essentially distinct invariant solutions of the maximal rank wehave found in explicit form, or we reduced nding those solutions to solvingthe classical one-dimensional equations of the mathematical physics: theheat equation, the telegraph equation, the Tricomi equation, the generalizedDarboux equation, and other equations.

The author was supported by the Russian Foundation for Basic Research

(project no. 16-01-00446).

REFERENCES

1. Chirkunov Yu.A., Conformal invariance and new exact solutions of the elasto-statics equations, J. Math. Phys., 58, No. 3, 031502 (2017).

116


INVARIANT MODELING OF THE MODELOF THERMAL MOTION OF GAS

IN A RAREFIED SPACE

Chirkunov Yu.A.1, Pikmullina E.O.2

1Novosibirsk State University of Architecture and Civil Engineering,Novosibirsk, Russia; [email protected]

2Novosibirsk State Technical University, Novosibirsk, Russia;[email protected]

A model describing the thermal motion of gas in a rareed space isinvestigated in [1]. This model can be used in the study of the motion ofgas in outer space, and the processes occurring inside the tornado, andthe state of the medium behind the shock front of the wave after a veryintense explosion [24]. The multi-dimensional model was investigated in[57]. The two-dimensional model was investigated in [8]. Exact solutionsand conservation laws for the three-dimensional model were obtained in [9].For a given initial pressure distribution, a special choice of mass Lagrangevariables leads to a reduced system of dierential equations describing thismotion, in which the number of independent variables is one less than theoriginal system. This means that there is a stratication of a highly rareedgas with respect to pressure. Namely, in a strongly rareed space for eachgiven initial pressure distribution, at each instant of time all gas particlesare localized on a two-dimensional surface moving in this space. At eachpoint of this surface, the acceleration vector is collinear with its normalvector. The resulting system admits an innite Lie transformation group.All signicantly various submodels that are invariant with respect to thesubgroups of its eight-parameter subgroup generated by the transfer, ex-tension, rotation, and hyperbolic rotation operators (the Lorentz operator)are found. For invariant submodels of rank 1, the basic mechanical charac-teristics of the gas ow described by them are obtained. Conditions forthe existence of these submodels are given. For invariant submodels ofrank 2, integral equations describing these submodels are obtained. For somesubmodels, the problem of describing the gas ow from the initial locationof its particles and the distribution of their velocities has been investigated.

The reported study was funded by the Russian Foundation for Basic Research

according to the research project no. 16-01-00446 a.

117


REFERENCES

1. Ovsyannikov L.V., The PODMODELI program. Gas dynamics, J. Appl.Math. Mech., 58, No. 4, 601627 (1994).

2. Sedov L. I., Propagation of strong blast waves [in Russian], J. Appl. Math.Mech., 10, No. 2, 241250 (1946).

3. Taylor G., The formation of a blast wave by a very intense explosion. I.Theoretical discussion, Proc. R. Soc. Lond. A, 201, No. 1065, 159174 (1950).

4. von Neuman J., The point source solution, in: Bethe H.A., Fuchs K.,Hirschfelder J.O., Magee J. L., Peierls R. E., von Neumann J., Blast Wave,Los-Alamos Scientic Laboratory Report LA2000, 1958, pp. 2755.

5. Chirkunov Yu.A., The conservation laws and group properties of the equa-tions of gas dynamics with zero velocity of sound, J. Appl. Math. Mech., 73,No. 4, 421425 (2012).

6. Chirkunov Yu.A., Group Analysis of Linear and Quasi-Linear DierentialEquations [in Russian], NSUEM, Novosibirsk (2007).

7. Chirkunov Yu.A., Khabirov S.V., Elements of Symmetry Analysis of Dieren-tial Equations of Continuum Mechanics [in Russian], NSTU, Novosibirsk(2012).

8. Khabirov S.V., The plane isothermal motions of an ideal gas without expan-sions, J. Appl. Math. Mech., 78, No. 3, 287297 (2014).

9. Chirkunov Yu.A., Exact solutions of the system of the equations of thermalmotion of gas in the rareed space, Int. J. Non-Linear Mech., 83, 914 (2016).

118


ON LUZIN N-PROPERTYFOR SOBOLEV MAPPINGS

Ferone A.1, Korobkov M.V.2, Roviello A.3

1Second University of Naples, Caserta, Italy;[email protected]

2Sobolev Institute of Mathematics SB RAS, Novosibirsk, Russia;[email protected]

3Second University of Naples, Caserta, Italy;[email protected]

We say that mapping v : Rn → Rd satises the (τ, σ)-N-property, ifHσ(v(E)) = 0 whenever Hτ (E) = 0, where Hτ means the Hausdor mea-sure. We prove that every mapping v of Sobolev class W k

p (Rn,Rd) withkp > n satises (τ, σ)-N-property for every 0 < τ 6= τ∗ := n− (α− 1)p with

σ = σ(τ) :=

τ, if τ > τ∗,pτ

αp− n+ τ, if 0 < τ < τ∗.

We prove also, that for k > 1 and for the critical value τ = τ∗ thecorresponding (τ, σ)-N-property fails in general. Nevertheless, this (τ, σ)-N-property holds for τ = τ∗, if we assume in addition that the highestderivatives ∇kv belong to the Lorentz space Lp,1(Rn) instead of Lp.

We extend these results to the case of fractional Sobolev spaces and forthe Besov spaces as well. Also, we establish some Fubini type theorems forN -properties and discuss their applications to the MorseSard theorem andits recent extensions.

The most results are contained in the recent preprint [1]. The proofs ofthe most results are based on our previous joint papers with J. Bourgain(Princeton) and J. Kristensen (Oxford), see [2, 3].

Similar results were announced by G. Alberti, M. Csornyei, E. D'Aniello,and B. Kirchheim (private communication, see also [4]).

119


REFERENCES

1. Ferone A., Korobkov M.V., Roviello A., On Luzin N-property and uncertaintyprinciple for the Sobolev mappings, arxiv.org/abs/1706.04796.

2. Bourgain J., Korobkov M.V., Kristensen J., On the MorseSard propertyand level sets of Wn,1 Sobolev functions on Rn, J. Reine Angew. Math., 700,93112 (2015).

3. Hajlasz P., Korobkov M.V., Kristensen J., A bridge between DubovitskiiFederer theorems and the coarea formula, J. Funct. Anal., 272, No. 3, 12651295 (2017).

4. Alberti G., Generalized N-property and Sard theorem for Sobolev maps, AttiAccad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl.,23, No. 4, 477491 (2012).

120


ON MODELS OF CIRCULAR GENE NETWORKS

Golubyatnikov V.P.1,2, Kirillova N. E.2

1Sobolev Institute of Mathematics SB RAS, Novosibirsk, Russia;[email protected]

2Novosibirsk State University, Novosibirsk, Russia;[email protected]

We study 10-dimensional nonlinear dynamical system

dmj

dt= −kjmj + fj(pj−1),

dpjdt

= µjmj − νjpj , j = 1, . . . , 5, (1)

as a model of a circular gene network functioning. Here j−1 = 5 if j = 1, thenon-negative variables pj , mj denote concentrations of proteins and mRNA,the coecients kj , µj , νj are positive, and the functions fj(p) are assumedto be positive, smooth and monotonically decreasing. This corresponds tonegative feedbacks in the gene network. Similar 6-dimensional symmetricsystem (f1 = f2 = f3, kj = 1, ν1 = µ1 = ν2 = µ2 . . ., etc.) was introducedin [1], see also [2]; its 6D asymmetric version was studied in [3].

Theorem 1.1. The system (1) has a unique equilibrium point.

2. If this point is hyperbolic, then the system (1) has at least one cycle.Let (2) be 18-dimensional partially symmetric system analogous to (1),

i.e., fs = fs+3, ks = ks+3, µs = µs+3, νs = νs+3, s = 1, . . . , 6.Theorem 2.

1. The system (2) has a unique equilibrium point.

2. If this point is hyperbolic, then the system (2) has at least two cycles.The authors were supported by the Russian Foundation for Basic Research

(project no. 15-01-00745).

REFERENCES

1. Elowitz M.B., Leibler S., A synthetic oscillatory network of transcriptionalregulator existence, Nature, London, 403, 335338 (2000).

2. Kolesov A.Yu., Rozov N.Kh., Sadovnichii V.A., Periodic solutions of travel-ling-wave type in circular gene networks, Izv. Math., 80, No. 3, 523548(2016).

3. Ayupova N.B., Golubyatnikov V.P., Kazantsev M.V., On existence of a cyclein one asymmetric model of a molecular repressilator [in Russian], Sib. Zh.Vychisl. Mat., 20, No. 2, 121129 (2017).

121


BINARY CORRESPONDENCES ANDPROBLEMS OF CHEMICAL KINETICSWITH MANY-SHEETED SLOW SURFACE

Gutman A.E.1, Kononenko L. I.2

Sobolev Institute of Mathematics SB RAS,Novosibirsk State University, Novosibirsk, Russia;


Formally, a problem is a binary correspondence P = (A,B,C) withC ⊆ A × B. The sets A, B, and C are treated as the domain of data, thedomain of unknowns, and the condition of the problem P . The containment(a, b) ∈ C is written as P (a, b) and means that the unknown b ∈ B isa solution to P for data a ∈ A. This approach provides a simple andadequate formalization for components of problems, their properties andconstructions, makes it possible to formalize topological problems, theirparametrizations, and dependence of solutions on parameters (see [1]).

As an example, we consider a singularly perturbed system of ordinarydierential equations which describes a process of chemical kinetics withmany-sheeted slow surface (see [2]). Suppose that n ∈ N, 0 < ε0 ∈ R, X isa domain in Rn, F := C(X2 × R× [0, ε0], Rn). Let P be the problem withdata F 2 × [0, ε0], unknowns C1(R, X)2, and condition P

((f, g, ε), (x, y)

)⇔

x(t) = f(x(t), y(t), t, ε

), ε y(t) = g

(x(t), y(t), t, ε

)for all t ∈ R. The formal

inverse P−1 to the problem P is impractical and can be corrected by meansof composition with the auxiliary problem Q with data (Rm)3, unknownsC1(R, X)2, and condition Q

((t, α, β), (x, y)

)⇔ x(ti) = αi, x(ti) = βi, i ∈

1, . . . ,m. In [1], solvability of the composition problem P−1 Q is studiedin a particular case.

The second author was supported by the Russian Foundation for Basic

Research (project no. 15-01-00745).

REFERENCES

1. Gutman A.E., Kononenko L. I., Formalization of inverse problems andapplications to systems of equations with parameters, in: Geometric Analysisand Control Theory: Abstracts, Sobolev Institute of Mathematics SB RAS,Novosibirsk, 2016, pp. 4042.

2. Gol'dshtein V.M., Sobolev V.A., Qualitative Analysis of Singularly PerturbedSystems [in Russian], Institute of Mathematics, Novosibirsk (1988).

122


CRITERION FOR THE POTENTIALITYOF A VECTOR FIELD

IN THE SUB-RIEMANNIAN GEOMETRY

Isangulova D.V.

Sobolev Institute of Mathematics SB RAS, Novosibirsk State University,Novosibirsk, Russia; [email protected]

We consider a subsetM of RN with completely non-integrable horizontaldistribution H. Suppose C∞-smooth horizontal vector elds X1, . . . , Xn

form a basis of Hx at each point x ∈M . Complete the elds X1, . . . , Xn toa basis of the whole TM so that Xn+1, . . . , XN are formed by commutatorsof elds from H: Xk =

∑I e

kI (x)XI , where I = (i1, . . . , im) ∈ 1, . . . , nm,

XI = Xi1 . . . Xim , k = n+ 1, . . . , N , [Xi, Xj ] =∑k

cijk(x)Xk.

The subgradient of a function f is a horizontal vector eld ∇Lf =X1fX1 + · · · + XnfXn ∈ H. We solve the problem posed in [1]. Whatconditions should satisfy the functions a1, . . . , an so that the horizontal vec-tor eld a1X1 + · · ·+ anXn ∈ H is the subgradient of some function f?

Theorem. Let a1, . . . , an be C∞-smooth functions on a simply con-

nected domain Ω of sub-Riemannian manifold M . Horizontal vector eld

a1X1 + · · ·+ anXn ∈ H is a subgradient of some function if

Xiaj(x)−Xjai(x) =∑k

cijk(x)ak(x), x ∈ Ω, 1 6 i < j 6 N, (1)

where an+1, . . . , aN can be calculated as

ak(x) =∑I

ekI (x)Xi1 . . . Xim−1aim(x), x ∈ Ω, k = n+ 1, . . . , N. (2)

Note that N(N − 1)/2 conditions (1) are not always independent. Usu-ally they include N − n conditions (2). In the case of Carnot groups, thenumber of conditions can be further reduced: we must check no more than(n− 1)(2N − n)/2 conditions. In the talk we write out the criterion ofpotentiality on several examples of sub-Riemannian manifolds and Carnotgroups.

REFERENCES

1. Calin O., Chang D.C., Sub-Riemannian Geometry: General Theory andExamples, Encycl. Math. Appl., Vol. 126, Cambridge University Press, Cam-bridge (2009).

123


HARDY TYPE INEQUALITIES FOR MONOTONEAND QUASI-MONOTONE FUNCTIONS

Jain P.

South Asian University, New Delhi, India;pankaj.jain@@sau.ac.in

We shall discuss inequalities of the type

∞∫0

(Sφf)p

(x)w(x) dx ≤ C∞∫

0

fp(x)w(x) dx, 1 < p <∞,

and its reverse inequality, where Sφ is the Hardy type operator

Sφf(x) :=1

Φ(x)

x∫0

f(t)φ(t)dt, Φ(x) :=

x∫0

φ(t)dt,

φ is a suitable function. We shall deal with the cases when in these inequali-ties functions f are monotone as well as quasi-monotone. All these inequali-ties will be also discussed in the framework of grand Lebesgue spaces.

124


LINEAR CANONICAL TRANSFORM CONNECTEDWITH VARIOUS CONVOLUTIONS

Kanjilal S.

South Asian University, New Delhi, India;[email protected],

[email protected]

We consider various types of convolutions given as:

• (f ?1 g)(t) =∫Rf(x)g(t− x) dx;

• (f ?2 g)(t) =∫Rf(x)g(x− t) dx;

• (f ?3 g)(t) =∫Rf(x)g(t+ x) dx;

• (f ?4 g)(t) =∫Rf(x)g(−x− t) dx.

In connection with these convolutions, we shall talk about the corre-sponding Linear Canonical Transform (LCT).

The above-mentioned convolutions and LCT will be discussed for func-tions as well as distributions. Various properties in this regard will bederived.

125


ZAYED TYPE FRACTIONAL CONVOLUTIONSAND DISTRIBUTIONS

Kumar R.

University of Delhi, New Delhi, India;[email protected]

The present talk focuses on three types of fractional convolutions, two ofwhich were given by A. Zayed. Our rst consideration is to extend thefamous Young's inequality in the context of fractional convolutions. Thenfor these convolutions weighted iterated Lp-Lq inequalities will be discussed.Finally, the notion of fractional convolutions between functions and distri-butions and also between distributions will be considered and various cor-responding properties will be discussed.

126


ENTROPY SOLUTIONS OF GENUINELYNONLINEAR FORWARD-BACKWARDULTRA-PARABOLIC EQUATIONS

Kuznetsov I.V.1, Sazhenkov S.A.2

Lavrentyev Institute of Hydrodynamics SB RAS, Novosibirsk, Russia;[email protected], [email protected]

Let vector functions a(z) and ϕ(z) satisfy the following conditions.Conditions on a&ϕ. Let a(z) = (a1(z), . . . , am(z)), ai ∈ C2(R), i =

1, . . . ,m, a(0) = 0, ϕ(z) = (ϕ1(z), . . . , ϕd(z)), z ∈ R, ϕj ∈ C2(R), j =1, . . . , d, ϕ(0) = 0. At least one component of a is a non-monotonic function.

Moreover, a′ satises the genuine nonlinearity condition

mes

λ ∈ R :

m∑i=1

a′i(λ)ξi = 0

= 0

for every (ξ1, . . . , ξm) ∈ Sm−1.

Let GT1,...,Tm = Ω× (0, T1)× . . .× (0, Tm), Γ10 = Ω×t1 = 0× [0, T2]×

. . .× [0, Tm], Γ1T1

= Ω×t1 = T1× [0, T2]× . . .× [0, Tm], Ξ1 = Ω× (0, T2)×. . . × (0, Tm),. . . ,Γm0 = Ω × [0, T1] × . . . × [0, Tm−1] × tm = 0, ΓmTm =

Ω× [0, T1]× . . .× [0, Tm−1]×tm = Tm, Ξm = Ω× (0, T1)× . . .× (0, Tm−1),Γl = ∂GT1,...,Tm\(Γ1

0 ∪ Γ1T1∪ . . . ∪ Γm0 ∪ ΓmTm). A bounded domain Ω ⊂ Rd

(mes Ω <∞) has smooth boundary ∂Ω.Problem Π0. For arbitrary initial and nal conditions u

i0, u

iTi∈ L∞(Ξi)

∩W 1,20 (Ξi), i = 1, . . . ,m, the unknown function u : GT1,...,Tm → R satises

divta(u) + divxϕ(u) = ∆xu, (x, t) ∈ GT1,...,Tm ,

u|Γi0 ≈ u10, u|ΓiTi ≈ u

iTi , i = 1, . . . ,m, u|Γl = 0,

where x = (x1, . . . , xd), t = (t1, . . . , tm), the sign ≈ means the equality only

on a part of the boundary.

Under Conditions on a&ϕ and the additional restriction on C1-norm ofϕ′, we have proved the existence and the uniqueness of the entropy solutionto problem Π0. The existence of quasi-solutions is also discussed.

The authors were supported by the grant no. III.22.4.2.

127


DUALITY OF SOME GENERALIZEDGRAND LEBESGUE SPACES

Singh M.

University of Delhi, New Delhi, India;[email protected]

Let I = (0, 1). The grand Lebesgue space, denoted by Lp)(I), is thespace of all nite a.e. measurable functions f for which

‖f‖Lp)(I) := sup0<ε<p−1

ε∫I

|f(x)|p−εdx

1p−ε

<∞, 1 < p <∞.

These spaces are originated by Iwaniec and Sbordone [1] in 1992 and haveundergone several generalizations during the recent past. The most recentgeneralization is the fully measurable grand Lebesgue spaces. In this talkwe shall be dealing with the properties of these spaces, viz., duality andreexivity. Also we shall talk about grand BochnerLebesgue spaces andtheir properties.

REFERENCES

1. Iwaniec T., Sbordone C., On the integrability of the Jacobian under minimalhypotheses, Arch. Ration. Mech. Anal., 119, No. 2, 129143 (1992).

128


DISCRETIZATION OF TWO-DIMENSIONALGELFANDLEVITAN INTEGRAL EQUATION

Temirbekova L.N.

Abai Kazakh National Pedagogical University,Almaty, Republic of Kazakhstan; [email protected]

In this work, we consider numerical method of solving inverse problemsfor two-dimensional hyperbolic equations. The coecient inverse problemis numerically solved using the method of the two-dimensional GelfandLevitan integral equation [1]. To numerically solve the two-dimensionalGelfandLevitan integral equation, which is the Fredholm integral equationof rst kind, we use parallel algorithms [2].

Consider the following sequence of the direct problems [3]

u(k)tt = u(k)

xx + u(k)yy − q(x, y)u(k), x > 0, y ∈ (−π, π), t ∈ R, k ∈ Z,

u(k)|t=0 = 0, u(k)t |t=0 = h(y)δ(x),

u(k)|y=π = u(k)|y=−π.

The inverse problem is to restore continuously function q(x, y) by the nextadditional information

u(k)(0, y, t) = f (k)(y, t), y ∈ (−π, π), t > 0, k ∈ Z,

where R is a set of real numbers, Z is a set of integer numbers, δ is Diracdelta function, k is some xed number, and h(y) = eiky; f (k)(y, 0) = 0is necessary condition. The problem considered above is solved by two-dimensional GelfandLevitan integral equation of the rst kind, which hasthe form [14]

1

2

[f (k)(y, t+x)+f (k)(y, t−x)

]+

x∫−x

∞∑m=1

f (k)m (t−s)ω(m)(x, y, s)ds = 0, x > |t|.

The integral equation is numerically solved by parallel algorithms.The author was supported by the project Theory and numerical methods

of solving inverse and ill-posed problems of natural science of the Ministry of

Education and Science of Kazakhstan (project no. 1746/GF4).

129


REFERENCES

1. Gelfand I.M., Levitan B.M., On the determination of a dierential equationfrom its spectral function [in Russian], Izv. Akad. Nauk SSSR. Ser. Mat., 15,309360 (1951).

2. Kabanikhin S. I., Inverse and Ill-Posed Problems [in Russian], Siberian Scien-tic Publishers, Novosibirsk (2009).

3. Romanov V.G., Inverse Problems of Mathematical Physics [in Russian],Nauka, Moscow (1984).

4. Kabanikhin S. I., Satybaev A.D., Shishlenin M.A., Direct Methods of SolvingInverse Hyperbolic Problems, VSP/BRILL, the Netherlands (2004).

130


LOTKAVOLTERRA COMPETITION MODELSIN ADVECTIVE ENVIRONMENTS

Vasilyeva O. I.

Memorial University of Newfoundland, Grenfell Campus,Corner Brook, Canada; [email protected]

The ow speed in streams and rivers can change due to natural causesor human activities. Any change in ow speed can impact biological com-munities that are shaped not only by their biotic interactions but also bytheir response to water ow. Population dynamics of several competingspecies in such habitats can be described by spatial LotkaVolterra com-petition models given by reaction-diusion-advection equations subject toDanckwerts boundary conditions. We use the dominant eigenvalue of thediusion-advection operator to reduce the original models to the corre-sponding systems of ordinary dierential equations.

Our analysis illustrates that at relatively high ow speed, each species'intrinsic growth rate is the crucial factor that determines the outcome ofcompetition. In contrast, at low ow speeds the strength of interspeciccompetition determines community composition. We show that changesin ow speed can facilitate dierent types of coexistence among multiplespecies.

REFERENCES

1. Vasilyeva O., Lutscher F., How ow speed alters competitive outcome inadvective environments, Bull. Math. Biol., 74, No. 12, 29352958 (2012).

2. Vasilyeva O., Lutscher F., Competition of three species in an advective envi-ronment, Nonlinear Anal., Real World Appl., 13, No. 4, 17301748 (2012).

3. Vasilyeva O., Competition of multiple species in advective environments,Bull. Math. Biol., 79, No. 6, 12741294 (2017).

131


SOBOLEV SPACES AND QUASI-CONFORMALMAPPINGS IN (SUB)-RIEMANNIAN GEOMETRY

Vodopyanov S.K.

Sobolev Institute of Mathematics SB RAS, Novosibirsk, Russia;[email protected]

In this talk we describe metric properties of measurable mappings ofdomains in (sub)-Riemannian manifolds inducing isomorphisms on Sobolevspaces by the composition rule. We show that any such mapping can beredened on a set of measure zero to be quasi-isometric, when the exponentof summability is dierent from the Hausdor dimension of a (sub)-Riemannian manifold, or to coincide with a quasi-conformal mappingotherwise [1, 2].

We give a sketch for proving the followingTheorem. Let p ∈ [1,∞) \ n, and D,D′ be domains in the Rieman-

nian manifold M of the topological dimension n. A measurable mapping

ϕ : D → D′ induces an isomorphism of Sobolev spaces L1p by the composi-

tion rule if and only if ϕ coincides with some quasi-isometry Φ: D → Φ(D)almost everywhere, for which the domains Φ(D) andD′ are (1, p)-equivalent.

For proving a similar result on (sub)-Riemannian manifolds we provideadditional tools like dierentiability, etc. [3].

The author was supported by the Russian Foundation for Basic Research

(project no. 17-01-00801).

REFERENCES

1. Vodopyanov S.K., On admissible changes of variables for Sobolev functionson (sub)Riemannian manifolds, Dokl. Math., 93, No. 3, 318321 (2016).

2. Vodopyanov S.K., Composition operators on Sobolev spaces, in: Contem-porary Mathematics, 382, AMS, Ann Arbor, 2005, pp. 327342.

3. Vodopyanov S.K., Geometry of CarnotCaratheodory spaces and dieren-tiability of mappings, in: Contemporary Mathematics, 424, AMS, Ann Arbor,2007, pp. 247302.

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ÀÂÒÎÐÑÊÈÉ ÈÍÄÅÊÑ

Àáàøååâà Í.Ë. . . . . . . . . . . . . . . . . . 29Àáäóëèí È.Ì. . . . . . . . . . . . . . . . . . 103Àçèìîâ À.À. . . . . . . . . . . . . . . . . . . . . 93Àëåêñàíäðîâ Â.Ì. . . . . . . . . . . . . . . 30Àíèêîíîâ Ä.Ñ. . . . . . . . . . . . . . . . . . 31Àíèêîíîâ Þ.Å. . . . . . . . . . . . . . . . . . 32Àþïîâà Í.Á. . . . . . . . . . . . . . . . . . . . 32Áàëàêèíà Å.Þ. . . . . . . . . . . . . . . . . . 33Áàëäàíîâ Ä.Ø.. . . . . . . . . . . . . . . . .34Áåëîçóá Â.À. . . . . . . . . . . . . . . . . . . . 35Áåëîóñîâà Î.Å. . . . . . . . . . . . . . . . . 103Áèáåðäîðô Ý.À. . . . . . . . . . . . . . . . . 37Áëîõèí À.Ì. . . . . . . . . . . . . . . . . . . . 37Áîëäûðåâ À.Ñ. . . . . . . . . . . . . . . . . . 38Áîíäàðü Ë.Í.. . . . . . . . . . . . . . . . . . .39Áîðîâñêèõ À.Â.. . . . . . . . . . . . . . . . .40Âàñêåâè÷ Â.Ë. . . . . . . . . . . . . . . . . . . 41Âîëîêèòèí Å.Ï. . . . . . . . . . . . . . . . . 42Ãðåøíîâ À.Â.. . . . . . . . . . . . . . . . . . .43Ãðèíåñ Â. Ç. . . . . . . . . . . . . . . . . . . . . .19Äâîðíèöêèé Â.ß. . . . . . . . . . . . . . . .44Äåìèäåíêî Ã.Â. . . . . . . . . . . . . . . . . .45Äåíèñîâà Ò.Å. . . . . . . . . . . . . . . . . . . 46Äæåíàëèåâ Ì.Ò. . . . . . . . . . . . . . . . . 48Åãîðîâ À.À. . . . . . . . . . . . . . . . . . . . . 49Åãîðøèí À.Î. . . . . . . . . . . . . . . . . . . 50Åëåóîâ À.À. . . . . . . . . . . . . . . . . . . . . 51Åëåóîâà Ð.À. . . . . . . . . . . . . . . . . . . . 51Çàêàðèÿíîâà Í.Á. . . . . . . . . . . . . . . 51Çâÿãèí À.Â. . . . . . . . . . . . . . . . . . . . . 52Çâÿãèí Â. Ã. . . . . . . . . . . . . . . . . . 23, 54Çèêèðîâ Î.Ñ. . . . . . . . . . . . . . . . . . . . 55Çîëîòîòðóáîâà Ã.Î.. . . . . . . . . . . . .56Èáðàãèìîâà Í.À. . . . . . . . . . . . . . . . 57Èâàíîâ Â.Â. . . . . . . . . . . . . . . . . . . . . 59Èñêàêîâ Ñ.À. . . . . . . . . . . . . . . . . . . . 48

Êàáàíèõèí Ñ.È. . . . . . . . . . 60, 62, 70Êîæàíîâ À.È. . . . . . . . . . . . . . . . . . . 64Êîçëîâà Ì.Ã. . . . . . . . . . . . . . . . . . . . 35Êîíäàêîâà Å.À. . . . . . . . . . . . . . . . . 60Êîíîâàëîâà Ä.Ñ. . . . . . . . . . . . . . . . 65Êîíîíîâ À.Ä.. . . . . . . . . . . . . . . . . . .66Êîñöîâ Ý. Ã.. . . . . . . . . . . . . . . . . . . . .85Êðèâîðîòüêî Î.È. . . . . . . . . . . 60, 70Êóçíåöîâ Ï.À. . . . . . . . . . . . . . . . . . .68Ëàòûïîâ È.È. . . . . . . . . . . . . . . . . . . 69Ëàòûøåíêî Â.À. . . . . . . . . . . . . . . . 70Ëàøèíà Å.À. . . . . . . . . . . . . . . . . . . . 71Ëîìàêèí À.Â. . . . . . . . . . . . . . . . . . . 72Ëîìàêèíà Å.À. . . . . . . . . . . . . . . . . . 73Ëîìîâ À.À. . . . . . . . . . . . . . . . . . 74, 75Ëóêèíà Ã.À. . . . . . . . . . . . . . . . . . . . . 76Ëóêüÿíåíêî Â.À. . . . . . . . . . . . . . . . 35Ìàëþòèíà Ì.Â. . . . . . . . . . . . . . . . . 77Ìàìîíòîâ À.Å. . . . . . . . . . . . . . . . . . 78Ìàòâååâà È.È.. . . . . . . . . . . . . . . . . .79Ìèðåíêîâ Â.Å. . . . . . . . . . . . . . . . . . 80Íåùàäèì Ì.Â. . . . . . . . . . . . . . . . . . 32Íèêèòåíêî Å.Â. . . . . . . . . . . . . . . . . 82Íóðñåèòîâ Ä.Á.. . . . . . . . . . . . . . . . .93Îðëîâ Ñ.Ñ. . . . . . . . . . . . . . . . . . 77, 84Ïèìàíîâ Ä.Î. . . . . . . . . . . . . . . . . . . 85Ïëàòîíîâà K.C. . . . . . . . . . . . . . . . . 40Ïîñòíîâ Ñ.Ñ. . . . . . . . . . . . . . . . . . . . 86Ïðîêóäèí Ä.À. . . . . . . . . . . . . . . . . . 78Ðàìàçàíîâ Ì.Ä. . . . . . . . . . . . . . . . . 88Ðàìàçàíîâ Ì.È. . . . . . . . . . . . . . . . . 48Ñàáèòîâ Ê.Á. . . . . . . . . . . . . . . . . . . . 89Ñåäîâà Í.Î. . . . . . . . . . . . . . . . . . . . . 91Ñåðîâàéñêèé Ñ.ß. . . . . . . . . . . . . . . 93Ñèäîðîâ Ñ.Í. . . . . . . . . . . . . . . . . . . .89Ñêâîðöîâà Ì.À. . . . . . . . . . . . . . . . . 95

133

Òàëûøåâ À.À. . . . . . . . . . . . . . . . . . . 96Òåëåøåâà Ë.À.. . . . . . . . . . . . . . . . . .97Òóðáèí Ì.Â. . . . . . . . . . . . . . . . . 54, 99Óâàðîâà È.À. . . . . . . . . . . . . . . . . . . . 98Óñòþæàíèíîâà À.Ñ. . . . . . . . . . . . . 99Ôàäååâ Ñ.È. . . . . . . . . . . . . . . . . . . . . 85Ôåäîðîâ Â.Å. . . . . . . . . . . . . . . . . . . 100Ôåäîñååâ À.Â. . . . . . . . . . . . . . . . . . . 75Õàçîâà Þ.À. . . . . . . . . . . . . . . . . . . 102×àíûøåâ À.È. . . . . . . . . . . . . . . . . 103×å÷êèíà À. Ã. . . . . . . . . . . . . . . . . . . . 26×óåøåâà Í.À. . . . . . . . . . . . . . . . . . 104×óìàêîâ Ã.À. . . . . . . . . . . . . . . . . . . . 71×óìàêîâà Í.À. . . . . . . . . . . . . . . . . . 71Øàäðèíà Í.Í. . . . . . . . . . . . . . . . . 105Øàìàíàåâ Ï.À. . . . . . . . . . . . . . . . 106Øåìåòîâà Â.Â. . . . . . . . . . . . . . . . . . 84Øèøêàíîâà À.À. . . . . . . . . . . . . . . 108Øèøëåíèí Ì.À. . . . . . . . . . . . . . . . 62Øîëïàíáàåâ Á.Á. . . . . . . . . . . . . . . .62Ùåðáàêîâ Â.Â. . . . . . . . . . . . . . . . . 109Ûñêàê Ò.Ê. . . . . . . . . . . . . . . . . . . . . 110ßçîâöåâà Î.Ñ. . . . . . . . . . . . . . . . . . 106Abiev N.A.. . . . . . . . . . . . . . . . . . . . .111

Antontsev S.N. . . . . . . . . . . . . . . . . . 112

Apanasov B.N. . . . . . . . . . . . . . . . . . 113

Assanova A.T. . . . . . . . . . . . . . . . . . 114

Chirkunov Yu.A. . . . . . . . . . . 116, 117

Feireisl E. . . . . . . . . . . . . . . . . . . . . . . . 24

Ferone A. . . . . . . . . . . . . . . . . . . . . . . 119

Golubyatnikov V.P. . . . . . . . . . . . . 121

Gutman A.E. . . . . . . . . . . . . . . . . . . 122

Isangulova D.V. . . . . . . . . . . . . . . . . 123

Jain P. . . . . . . . . . . . . . . . . . . . . . . . . . 124

Kanjilal S. . . . . . . . . . . . . . . . . . . . . . .125

Kirillova N.E. . . . . . . . . . . . . . . . . . . 121

Kononenko L. I. . . . . . . . . . . . . . . . . 122

Korobkov M.V. . . . . . . . . . . . . . . . . 119

Kumar R. . . . . . . . . . . . . . . . . . . . . . . 126

Kuznetsov I. V. . . . . . . . . . . . . . . . . . 127

Pikmullina E.O. . . . . . . . . . . . . . . . . 117

Roviello A. . . . . . . . . . . . . . . . . . . . . . 119

Sazhenkov S.A. . . . . . . . . . . . . . . . . 127

Singh M. . . . . . . . . . . . . . . . . . . . . . . . 128

Temirbekova L.N. . . . . . . . . . . . . . . 129

Vasilyeva O. I. . . . . . . . . . . . . . . . . . . 131

Vodopyanov S.K.. . . . . . . . . . . . . . .132

134

Íàó÷íîå èçäàíèå

ÑÎÁÎËÅÂÑÊÈÅ ×ÒÅÍÈß

Ìåæäóíàðîäíàÿ øêîëà-êîíôåðåíöèÿÍîâîñèáèðñê, Ðîññèÿ, 2023 àâãóñòà 2017 ã.

ÒÅÇÈÑÛ ÄÎÊËÀÄÎÂ

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