Chao-Jiang Xu and Tong Yang- Local Existence with Physical Vacuum Boundary Condition to Euler...

8/3/2019 Chao-Jiang Xu and Tong Yang- Local Existence with Physical Vacuum Boundary Condition to Euler Equations with Da

1/12

LOCAL EXISTENCE WITH PHYSICAL VACUUM BOUNDARY

CONDITION TO EULER EQUATIONS WITH DAMPING

CHAO-JIANG XU AND TONG YANG

Abstract In this paper, we consider the local existence of solutions to Euler equations

with linear damping under the assumption of physical vacuum boundary condition. By

using the transformation introduced in [13] to capture the singularity of the boundary,we prove a local existence theorem on a perturbation of a planar wave solution by

using Littlewood-Paley theory and justifies the transformation introduced in [13] in arigorous setting.

Key words Euler equations, physical vacuum boundary condition, Littlewood-

Paley theory, local existence.A.M.S. Classification 35L67, 35L65, 35L05.

1. Introduction

In this paper, we are interested in the time evolution of a gas connecting to vacuumwith physical boundary condition. By assuming that the governed equations for the gasdynamics are Euler equations with linear damping, cf. [16] for physical interpretation,one can see that the system fails to be strictly hyperbolic at the vacuum boundarybecause the characteristics of different families coincide. As discussed in the previousworks, cf. [5, 11, 12, 13], the canonical vacuum boundary behavior is the case whenthe space derivative of the enthalpy is bounded but not zero. In this case, the pressurehas its non-zero finite effect on the evolution of the vacuum boundary. However, forthis canonical (physical) case, the system becomes singular in the sense that it can notbe symmetrizable with regular coefficients so that the local existence theory for theclassical hyperbolic systems can not be applied. Furthermore, the linearized equationat the boundary gives a Keyldish type equation for which general local existence theoryis still not known. Notice that this linearized equation is quite different from the oneconsidered in [18] for weakly hyperbolic equation which is of Tricomi type. To capturethis singularity in the nonlinear settting, a transformation was introduced in [13] andsome local existence results for bounded domain were also discussed. The transformedequation is a second order nonlinear wave equation of an unknown function ( ) with

coefficients as functions of 1

( ) and (0 ) 0. Along the vacuum boundary,the physical boundary condition implies that the coefficients are functions of y(0 )which are bounded and away from zero. Hence, the wave equation has no singularityor degeneracy. However, its coefficients have the above special form so that the localexistence theory developed for the classical nonlinear wave equation can not be applieddirectly, [8, 9]. There are other works on this system with vacuum, please refer to[6, 10] ect. and reference therein.

TO APPEAR IN J. DIFF. EQU.1


2/12

2 XU AND YANG

Even though a transformation to capture the singularity in the physical boundarycondition at vacuum interface is introduced in [13], the energy method presented theremay not give a rigorous proof of the existence theory, especially in the general setting.It is because the coefficients in the reduced wave equation which are power functionsof 1 correspond to the fractional differentiations of . Under this consideration,

we think the application of Littlewood-Paley theory based on Fourier theory is moreappropriate. Therefore, as the first step in this direction, in this paper we will studythe local existence of solutions satisfying the physical boundary condition when theinitial data is a small perturbation of a planar wave solution where the enthalpy islinear in the space variable, [11]. By applying the Littlewood-Paley theory, we obtainthe solution local in time with the prescribed physical boundary condition.

Precisely, we consider the one dimensional compressible Euler equations for isentropicflow with damping in Eulerian coordinates

t + ( )x = 0

t + x + ( )x = (1.1)

where , and ( ) are density, velocity and pressure respectively. And the linearfrictional coefficient is normalized to 1. When the initial density function containsvacuum, the vacuum boundary is defined as

= {( ) | ( ) 0} {( ) | ( ) = 0}Since the second equation in (1.1) can be rewritten as

t + x + x = with being the enthalpy, one can see that the term x represents the effect of thepressure on the particle path, in particular, on the vacuum boundary. It is shownin [12, 15, 19] that there is no global existence of regular solutions satisfying x 0along the vacuum boundary. That is, in general, is not

1

crossing the vacuumboundary. Hence, the canonical behavior of the vacuum boundary should satisfy thecondition x = 0 and is bounded. This special feature of the solution can be illustratedby the stationary solutions and some self-similar solutions, also for different physicalsystems, such as Euler-Poisson equations for gaseous stars and Navier-Stokes equations,cf. [5, 11, 13, 17]. Notice that the charateristics of Euler equations is , with

=

( ). And for isentropic polytropy gas, =c2

1 , where 1 is the adiabaticconstant. Hence the characteristics are singular with infinite space derivative at thevacuum boundary if physical boundary condition is assumed. This singularity yieldsthe smooth reflection of the characteristic curves on the vacuum boundary and thencauses analytical difficulty.

Another way to view the canonical boundary condition comes from the study ofporous media equation. It is known that the Euler equations with linear dampingbehave like the porous media equation at least away from vacuum when ,cf.[7] and some corresponding results in the weak sense with vacuum which will notbe discussed here. For the porous media equation, the free boundary of the supportof the solution has a canonical behavior which would be the same as or similar tothe one described above for Euler equations with damping. However, there is stillno satisfatory results on the change of solution behavior along the vacuum boundary


3/12

LOCAL EXISTENCE 3

even though the corresponding waiting time problem for porous media equation is wellunderstood, cf.[1].

In this paper, we will concentrate on the Euler equations with linear damping whenthe initial data is a small perturbation of a planar wave in one dimensional space.Since our concern is the behavior of the solution related to vacuum and any shock

wave vanishes at vacuum [14], it is reasonable to consider our problem without shockwaves. In fact, any shock wave appears initially or in finite time will decay to zeroexponentially in time because of the dissipation from the linear damping. By usingthe special property of the one dimensional gas dynamics, we can rewrite the system(1.1) by using Lagrangian coordinates to make all the particle paths, in particular thevacuum boundary, as straight lines. (1.1) in Lagrangian coordinates takes the form

t = 0t + ( ) = (1.2)

where = 1

is the specific volume and =x

0( ) . Moreover, we assume that the

pressure function satisfies the -law, i.e., ( ) = 2 , 1. Notice that the physicalsingularity, x = 0 but bounded, along the vacuum boundary in Eulerian coordinatescorresponds to 0 | ( )| in the Lagrangian coordinates.

In order to capture this singularity in the solution and symmetrize the system (1.2),the following coordinate transformation was introduced in [13],

=21

Here, we assume that the initial density function 0( ) = 0 for 0 in the Euleriancoordinates. Then the system (1.2) can be rewritten as

( )t + y = 0

t + ( )y =

0 0(1.3)

where ( ) =21

12 , and

=( 1) ( 21 )+12 = ( 1 ) +11

for some positive constant . Without any ambiguity and up to a scaling, can bechosen to be 1 and we still denote the independent variable by for simplicity ofnotation. Notice that near the vacuum boundary, both ( )y and are bounded awayfrom zero under the physical boundary condition.

Therefore, the vacuum problem considered can be formulated into the followingboundary value problem:

( 1 t)t ( y)y + 1 t = 0(1.4)( t)|t=0 = ( 0 1)(1.5)

(0 ) = 0(1.6)

0 1 1 ( ) 2(1.7)with = ( 1 ( )) 1 and compatibilities conditions 2y 0(0) = 0 =0 1 2


4/12

4 XU AND YANG

It is easy to see that the above equation has a special linear unbounded solution for 0 given by ( ) = 0 with constant 0 0. This solution is also obtained in

[11] together with other self-similar solutions with physical boundary condition. Tojustify the above transformation for local existence purpose, we will consider the localexistence of solution when the initial data is a small perturbation to the above special

solution.That is, the initial data is assumed to be

0( ) = ( 0 + 0( )) 1( ) = 1( )

for some constant 0 0. And the solution is of following type

( ) = ( 0 + ( ))

Then the problem on (1.4)-(1.7) in this setting becomes

tt ( 2 y)y + ( 0 + )31 2y + 2 2

y 2t

0 ++ t = 0(1.8)

( t)

|t=0 = ( 0 1)

s(R+)

s1(R+)(1.9)

(0 ) = 0 0(1.10)

L([0,T]R +)

1

20(1.11)

with = ( 0 + ( )) and 1.

For this problem, we have the following main theorem in this paper.

Theorem 1.1. Suppose that, for some 0 0, we have

(1.12) Supp 0 Supp 1 [ 0 +[ 0L( R +) 1

40

and 32 . Then there exists 00 0 such that the problems (1.8)-(1.11) has a

unique solution ([0 ] s(R+)) 0,1([0 ] s1(R+))

Notice that the case when 0 = 0 is more difficult and will not be discussed here.Since the solution is regular up to the vacuum boundary and the density function inpositive except on the vacuum boundary, the result in Theorem 1.1 can be reducedstraightforwardly to the solution to the Equations (1.1).

Notice that here the initial perturbation is in a compact subset in (0 ) and thelocal time existence is proved before the perturbation influence the propagation of theboundary. Therefore, it is interesting and important to consider how the behaviorof the boundary changes in later time due to the perturbation. But this is not in

the scope of this paper and will be pursued by the authors in the future. For this,the transformation introduced in [13] could still be useful. Furthermore, the physicalboundary condition holds also for multi-dimensional space by considering the stationarysolutions, [5]. Hence, the evolution of the vacuum interface in multi-dimensional spacecan also be considered with more difficulty because there is no Lagrangian coordinatesto fix the vacuum interface.

The rest of the paper is arranged as follows. In Section 2, we shall briefly includethe Littlewood-Paley theory for the proof of local existence. The proof of Theorem


5/12

LOCAL EXISTENCE 5

1.1 is given in Section 3 where a linearized system is analyzed to yield a sequence ofsolutions being convergent to the one in Theorem 1.1.

2. Littlewood-Paley theory

In this section, we will recall some elementary properties of Littlewood -Paley theoryfor the Sobolev spaces, for the details please refer to [2, 3, 4]. Set

s(Rd) = { S; (1 + | |2)s/2 2(Rd)}with the norm Hs = (1 + | |2)s/2 L2. We consider now a dyadic decompositionofRd. For 1 a fixed constant, and N+, we set(2.1) Cp = { Rd; 12p | | 2p+1}and C1 = (0 ) = { Rd; | | }, then {Cp}+1 is a uniformly finite recover ofR

d, that means, if | | 1 = 2(1 + 2 log2 ) + 2, we have Cq Cp = .We can also construct two functions

0 (R

d), with Supp

C1 Supp

C0,

such that for any Rd and 0,

( ) +

p=0

(2p ) = 1 ( ) +N01p=0

(2p ) = (2N0 )

Then one can define the following operators of localization in Fourier space, for S(Rd),

p = p = F1( (2p)()) = 2pd

R

d

(2p ) ( ) for Nand

1 =

1 =F1( (

)(

))

where = F( ) denotes the Fourier transformation of , and = F1( ). It is evidentthat p S for any S, Supp p Cp, and =

p=1 p, in sense ofS.

Since Supp p Cp, Paley-Wienner-Schwartz theorem implies that p and theSobolev space can be characterized as follows.

Lemma 2.1. For 0, the following properties are equivalent.(a) s(Rd);(b) =

p=1 p in S, Supp p Cp and pL2 p2ps { p} 2;

(c) =

p=1 p in S, Supp p (0 12p) and pL2 p2ps { p} 2;(d) =

p=1 p in S, p and for any Nd | | [ ] + 1,

pL2 p,2p(s||) { p,}pN 2Remark : The equivalence of (a) and (b) holds for all R.For the estimate, we need the following lemma.

Lemma 2.2. Suppose that (Rd) Supp (0 ), then (Rd), and forany Nd there exist ( ) 0 such that(2.2) L ( ) || L


6/12

6 XU AND YANG

For some 0 large enough, (0 4 2N0) is a very small ball. Set

C0 = C0 + (0 4 2N0)then {C p} = {2pC 0} has the same properties as {Cp}. We define

q = 1pqN0 p u = q ( q ) q ( ) = |pq|


7/12

LOCAL EXISTENCE 7

On the other hand, there exists 0 such that

k( a y ) ak( y ) =

|kk|N0(k( k( )k y ) k( )k(k( y )))

and

k( k( )k y ) k( )k(k( y ))( )= 2dk

(2k( ))( k( ) ( ) k( )( ))k y ( )

Hence

k( k( )k y ) k( )k(k( y ))L2 2k ( )L1 Lk y L2 N0 Hs Hs1

This completes the proof of the lemma.

3. Proof of the TheoremIn this section, we are going to prove the local existence of solution stated in Theorem

1.1. The proof is based on the study of a linearized problem. We want to construct aconvergent sequence of solutions to the linearized problem and show that the limit isthe solution to the nonlinear problem (1.8)- (1.11) with the property stated in Theorem1.1.

Under the hypothesis of theorem 1.1, we study now the sequence of functions { n}nN

defined inductively as follows.

1 = 0(3.1)n+1tt

(( n)2 n+1y )y =

n(3.2)

( n+1 n+1t )|t=0 = ( 0 1)(3.3)with

n( ) = ( 0 +n( ))

n( ) = ( 0 + n)31( n)2y ++ 2

( n)2 ny +nt

0 + nnt nt

For 0 10 0 and

0 = 1 02Hs + 12Hs11/2

with 1 = 2(8 0) if 0

2, and 1 = 2(2 0)

if 0 2, we define

s,T1 = | 0([0 1]; s(R+)) 0,1([0 1]; s1(R+))

||| |||Xs,T1 = 2L([0,T1];Hs( R +)) + t2L([0,T1];Hs1(R +))

1/2 0;

L(R +[0,T1])

1

20 Supp {( ) R+ [0 1]; + 0 0}

We will prove the following theorem.


8/12

8 XU AND YANG

Theorem 3.1. (a) For any 3 2 0 0 0 1 0 ( 0 ), if n s,T1,then the Cauchy problem (3.2)-(3.3) has a solution

n+1 0([0 1]; s(R+)) 0,1([0 1]; s1(R+))with

Supp n+1 {( ) [ +[[0 1]; + 0 0}(b) For any 3 2, there exists 0 1

0 0, such that if

n s,T1, thenthe solution n+1 of Cauchy problem (3.2)-(3.3) belongs to s,T1, that means that thesequence { n} is well-defined and uniformly bounded in s,T1.

(c) There exists 0 2 1 such that the sequence { n} is a Cauchy sequence ins1,T2.

Proof. First for part (a), since 0 1 0 ( 0 ), if n s,T1 , thenSupp n {( ) [ +[[0 1]; + 0 0}

We have thatn 0([0 1]; s(R)) 0,1([0 1]; s1(R))

andn 0([0 1]; s1(R)) 0,1([0 1]; s2(R))

Thus, the existence theorem for linear Cauchy problem gives the existence of solutionto (3.2) and (3.3), cf. [9]

n+1 0([0 1]; s(R)) 0,1([0 1]; s1(R))Moreover for n s,T1, we have n( ) = 0 n( ) = 0 0 = 1 = 0 for all( ) R+ [0 1]; + 0 0, so that in this domain, n+1 is the solution ofproblem

n+1tt 20 n+1yy = 0 ( t)|t=0 = (0 0)

Then n+1 = 0 in this domain and this gives part (a).We now turn to part (b). For 0 0, take (R) ( ) = 1y if ; ( ) =

2 if 2. We suppose always 0 1 0 ( 0 ). For s,T1 , we set

( ) = ( 0 + ( ))

( ) = ( 0 + )31()2y ++ 2

()2 y +2t

0 + t

Remark that ( ) y =1

y

y, since y( ) = 0 if

. Then by using Theorem 2.4, wehave

0([0 1]; s(R)) 0,1([0 1]; s1(R)) 0([0 1]; s1(R)) 0,1([0 1]; s2(R))

And

L([0,T1];Hs1(R )) 2 [s]+10 ; L([0,T1];Hs( R )) 2 [s]0(3.4)


9/12

LOCAL EXISTENCE 9

with constant 2 depends only on 0 0 and . We consider now the following linearproblems

tt (()2 y)y = (3.5)( t)|t=0 = ( 0 1)(3.6)

In fact, part (b) is equivalent to the following claim.

Claim: Suppose that

Supp 0 Supp 1 [ 0 +[ 0L(R +)

1

40

and 32 . There exists 1 0 depending on 0 0 0, such that, for any s,T1 ,the solution of problem (3.5)-(3.6) is also in s,T1.

For the above claim, we only need to prove the following estimate for the solutionof problem (3.5)-(3.6).

(3.7) 2L([0,T1];Hs(R )) + t2L([0,T1];Hs1( R )) 20By using Sobolev embedding theorem, Lipschitz estimate and boundedness of 0,we get immediately, for 1 0 (4 0 s),

L(R [0,T1]) 1

20

To apply the Lemma 2.1, we need the following estimate,

(3.8) k t2L([0,T1];L2( R )) + k y2L([0,T1];L2( R )) 2k22k(s1)

with

2k 20 for N.

Since tt

([0 1]; s2(R)), by applying k to the equation (3.5) and integratingits product with k t over ( ) in R [0 ], we have,

1

2

R

|k t|2( ) + 12

R

2|k y|2( ) = 12

R

|k 1|2( )

+1

2

R

2|k( 0)y|2( ) +t

0

R

k( )k( t)

+

t0

R

t|k( y)|2 t

0

R

[k 2] yk( ty)

Using Cauchy-Schwarz inequality, we have

k t2

L2(R )( ) + k y2

L2(R )( ) 1

42

1 (k 12

L2(R ) + k( 0)y2

L2(R ))

+ 2121k( )2L([0,T1];L2(R )) + 21 21 2kk([k 2] y)2L([0,T1];L2(R ))

+1

2k( )t2L([0,T1];L2(R )) +

1

41

21tL(R [0,T1])k y2L([0,T1];L2(R ))

where k = |kk|N1 k, and k k = k. We havetL(

R[0,T1]) (2 0)21 tL(R [0,T1]) (2 0)21 s tL([0,T1];Hs1(R ))


10/12

10 XU AND YANG

where 1 1 2. By choosing 0 1 small enough satisfying

121 (2 0)

21s tL([0,T1];Hs1(R )) 1 21 (2 0)21 s 0 2

we have

k t2L([0,T1];L2(R )) + k y2L([0,T1];L2(R )) 1221 (k 12L2(

R) + k( 0)y2L2(

R))

+2 2121k( )2L([0,T1];L2( R )) + 2 21 21 22kk([k 2] y)2L([0,T1];L2(R ))

Hence, (2.7) and (3.4) yield

k( )L([0,T1];L2(R )) k2k(s1) L([0,T1];Hs1( R )) 2 [s]+10 k2k(s1);

k([k

2] y)L([0,T1];L2(R )) k2ks2L([0,T1];Hs(R )) L([0,T1];Hs(R ))

2[s]

0 k2ks

L([0,T1];Hs(R ))with

2k 1. By choosing 0 1 1 2 [s]0

2 4 in the above estimate , we

complete the proof of the claim and then obtain the part (b).Finally, we want to prove part (c) of Theorem 3.1. Let { n} be a sequence of

functions defined by (3.1)-(3.3), we prove that there exists 0 2 1 such that itis a Cauchy sequence in 0([0 2];

s1(R)) 0,1([0 2]; s2(R)). In fact we willprove the following estimate, for any N,(3.9) n+1 n2L([0,T2];Hs1(R )) + n+1t nt 2L([0,T2];Hs2(R )) 2n 20

Set

n+1

=

n+1

n

N

, we haven+1tt (( n)2 n+1y )y = ( n n1) ((( n)2 ( n1)2) ny )y

( n+1 n+1t )|t=0 = (0 0)where n+1 n n1 s,T1, and

( n)2 ( n1)2 = 1( n n1) n;n n1 = 2( n n1 n n1) n

+ 3(n n1 n n1) nt + 4(

n n1 n n1) ny

with jL([0,T1];Hs1(R )) ( 0) = 1 4For [0 ] 0 1,

t0

R

k(n n1)k n+1t

( 0) k2k(s2)k n+1t L([0,T];L2( R )) nL([0,T];Hs1(R )) + nt L([0,T];Hs2( R ))


11/12

LOCAL EXISTENCE 11

and t

0

R

k(((n)2 ( n1)2) ny )yk n+1t

t

0 Rk((

n)2

( n1)2) ny k

n+1ty

( 0) k2k(s2)k n+1t L([0,T];L2(R )) nL([0,T];Hs1(R ))

By using (2.6), we havet

0

R

[k (

n)2] n+1y

kn+1ty

( 0) k2k(s2)k n+1t L([0,T];L2( R )) n+1L([0,T];Hs1(R ))

Then, we get

k n+1t 2L([0,T];L2) + k n+1y 2L([0,T];L2)

42

( 0)2 2

k22k(s

2)

(n

t 2

L([0,T];Hs2(R )) + n

y2

L([0,T];Hs2(R )))+4 2 ( 0)

2 2k22k(s2) n+12L([0,T];Hs1(

R))

By multipling this inequality by 22k(s2) and summing over , we have for 4 2 ( 0)2 1 2,

n+1t 2L([0,T];Hs2(R

)) + n+1y 2L([0,T];Hs2(R

))

8 2 ( 0)2( nt 2L([0,T];Hs2(R

)) + ny 2L([0,T];Hs2(R

)))

Now by choosing 8 2 ( 0)2 1 2, we have (3.9). This completes the proof of

Theorem 3.1.

Now the proof for Theorem 1.1 can be stated as a consequence of Theorem 3.1 asfollows. Since the sequence { n} is a Cauchy sequence in s1,T2 and bounded in s,T2 ,it is also the Cauchy sequence in s,T2 for all

by interpolation. Then the limitis in 0([0 2];

s(R+)) 0,1([0 2]; s1(R+)). Since 3 2, is a solution of equation (1.8) and this yields Theorem 1.1.

Acknowledgment: The research was supported CityU Direct Allocation Grant #7100198. And the authors would like to thank the referees insightful idea on improvingthe theorem in this paper.

References

[1] D. G. Aronson, L. A. Caffarelli and S. Kamin, How an initial stationary interface begins to movein porous media flow, SIAM J. Math., Anal., 14(1983), 639-658.

[2] J.-M. Bony, Calcul symbolique et propagation des singularites pour les equations aux derivees

partielles non lineaires, Annales de l Ecole Normale Superieure, 14, 1981, pages 209246.[3] J.-Y. Chemin, Fluides parfaits incompressibles, Asterisque, 230 (1995).[4] J.-Y. Chemin, B. Desjardins, I. Gallagher, E. Grenier, Fluids with anisotropic viscosity. Special

issue for R. Temams 60th birthday. M2AN Math. Model. Numer. Anal. 34, (2000), no. 2, 315-335


12/12

12 XU AND YANG

[5] Yinbin Deng, Tai-Ping Liu, Tong Yang and Zheng-an Yao, Solutions with vacuum of Euler-Poissonequations, Arch. Rat. Mech. Anal., 164(2002), 3, 261-285.

[6] M. Grassin and D. Serre, Global smooth solutions to Euler equations for an isentropic perfectgas, C.R. Acad. Sci. Paris Ser. I Math, 325(1997), No. 7, 721-726.

[7] L. Hsiao and T.-P. Liu, Convergence to nonlinear diffusion waves for solutions of a system ofhyperbolic conservation laws with damping, Comm. Math. Phys.143 (1992), 599-605.

[8] L. Hormander, Lectures on Nonlinear Hyperbolic Differential Equations, 1997, Springer-Verlag.[9] T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Rational.

Mech. Anal., 58 (1975), 181-205.[10] L.W. Lin, On the vacuum state for the Equations of Isentropic Gas Dynamics, J. of Math. Anal.

Appl., Vol. 121, No. 2 (1987) pp. 406425.[11] T.-P. Liu, Compressible flow with damping and vacuum, Japan J. Appl. Math., Vol. 13, No. 1,

25-32.[12] T.-P. Liu and T. Yang, Compressible Euler equations with vacuum, Journal of Differential Equa-

tions, Vol. 140, No. 2, 1997, 223-237.[13] T.-P. Liu and T. Yang, Compressible flow with vacuum and physical singularity, Methods and

Applications of Analysis, Vol. 7, No. 3, 495-510, 2000.[14] T.-P. Liu and J. Smoller, On the vacuum state for isentropic gas dynamics equations, Advances

in Math., 1(1980), 345-359.

[15] T. Makino, Blowing up solutions of the Euler-Poisson equation for the evolution of gaseous stars,Transport Theory and Statistical Physics 21 (1992), 615624.

[16] T. Nishida, Nonlinear hyperbolic equations and related topics in fluid dynamics, Publ. Math.DOrsay, (1978), 46-53.

[17] M. Okada, Free boundary value problems for the equation of one-dimensional motion of viscousgas, Japan J. Appl. Math., 6, No.1, 161-177, 1989.

[18] O.A. Oleinik, On the Cauchy problem for weakly hyperbolic equations, Comm. Pure Appl. Math.,Vol., XXIII, (1970), 569-586.

[19] Z. Xin, Blow-up of smooth solutions to the compressible Navier-Stokes equations with compactdensity, Comm. Pure Appl. math., 51 (1998), 229-240.

Chao-Jiang Xu, Universite de Rouen, Laboratoire de Mathematiques, 76821 Mont-

Saint-Aignan, France

E-mail address: [email protected]

Tong Yang, Department of Mathematics, City University of Hong Kong, Kowloon,

Hong Kong

E-mail address: [email protected]

Chao-Jiang Xu and Tong Yang- Local Existence with Physical Vacuum Boundary Condition to Euler...

Documents

Transcript of Chao-Jiang Xu and Tong Yang- Local Existence with Physical Vacuum Boundary Condition to Euler...