Stochastic Landau-Lifshitz-Gilbert Equation · Stochastic Landau-Lifshitz-Gilbert Equation Ben...

Stochastic Landau-Lifshitz-GilbertEquation

Ben Goldys (UNSW, Sydney)

Isaac Newton Institute, Cambridge, March, 2010

Ben Goldys (UNSW, Sydney) Stochastic Landau-Lifshitz-Gilbert Equation

co-authors

joint work with

Zdzisław Brzezniak (York University, UK)and

Terence Jegaraj (UNSW, Sydney)


Notations

D ⊂ Rd bounded open domain with smooth boundary, d ≤ 3

L2 = L2(

D,R3),

H1 = H1(

D,R3),

a · b, inner product in R3

a× b, vector product in R3


Physical background

We consider a ferromagnetic material fillinga domain D ⊂ Rd , d ≤ 3,

u(x) the magnetic moment at x ∈ D,

For temperatures not too high

|u(x)| = 1, x ∈ D


Energy functional I

Every configuration φ : D → R3, φ ∈ H1 of magnetic momentsminimizes the energy functional

E (φ) =a1

2

∫D|∇φ|2dx +

12

∫Rd|∇v |2dx −

∫D

H · φdx

∆v(x) = ∇ · φ(x), x ∈ Rd

φ(x) =

φ(x) if x ∈ D,0 if x /∈ D.

|φ(x)| = 1, x ∈ D.


Energy functional II

Landau-Lifschitz 1935, Gilbert 1955

a1

2

∫D|∇φ|2dx , exchange energy,

12

∫Rd|∇v |2dx , magnetostatic energy,

H- given external field.


Landau-Lifschitz-Gilbert equation

H (u) = −DuE (u) = a1∆u −∇v + H

∂u∂t = λ1u ×H (u)− λ2u × (u ×H (u)) on D

∂u∂n = 0 on ∂D

|u0(x)| = 1 on D

λ2 > 0

and from now onλ1 = λ2 = 1.


Connection with harmonic maps problem

E (φ) =12

∫D|∇φ|2dx

∂u∂t

= −u × (u ×∆u)

butu × (u ×∆u) = (u ·∆u)u − |u|2∆u,

|u|2 = 1 on D then

u · ∇u = 0, ⇒ u ·∆u = −|∇u|2

We obtain heat flow of harmonic maps:

∂u∂t

= ∆u + |∇u|2u


Previous works

A. Visintin 1985: weak existence, d ≤ 3,

Chen and Guo 1996, Ding and Guo 1998, Chen 2000, Harpes2004: existence and uniqueness of partially regular solutions,d = 2

C. Melcher 2005: existence of partially regular solutions, d = 3,

R. V. Kohn, M. G. Reznikoff, E. Vanden-Eijnden 2007, largedeviations

A. Desimone, R. V. Kohn, S. Müller, F. Otto 2002, thin filmapproximations

R. Moser 2004, thin film approximations, magnetic vortices


Thermal noise

E (φ) = · · · −∫

DH · φ

Néel 1946: H = noise.

H = hdW

h : D → R3, W Brownian Motion


Stochastic Landau-Lifschitz-Gilbert-Equation I

H (u) = −DuE (u) = ∆u −∇v + hdW

∂u∂t = u ×H (u)− u × (u ×H (u)) on D

∂u∂n = 0 on ∂D

|u0(x)| = 1 on D

F dW is a Stratonovitch integral:

F (u) dW =12

DF (u) · F (u)dt + FdW


Non-local term

∆v = ∇ · u, in Rd , u ∈ H1

Formally∇v = ∇∆u−1∇ · u

∇v = Pu, restricted to D

P =k|k |⊗ k|k |


Stochastic Landau-Lifschitz-Gilbert-Equation II

H (u) = ∆u − Pu + hdW

∂u∂t = u ×H (u)− u × (u ×H (u)) on D

∂u∂n = 0 on ∂D

|u0(x)| = 1 on D

(1)


Integration by parts

∆N Neumann Laplacian

D (∆N) =

u ∈ H2 :

∂u∂n

= 0, on ∂D

.

Lemma

If v ∈ H1 and u ∈ D (∆N) then∫D〈u ×∆N , v〉 dx =

∫D〈∇u, (∇v)× u〉dx .


Weak martingale solution

Definition

(Ω,F , (Ft )t≥0,P,W , u) is a solution to (2) if for every T > 0 andφ ∈ C∞

(D,R3)

u(·) ∈ C(

[0,T ]; H−1,2), P− a.s.

E supt≤T|∇u(t)|2L2 <∞,

|u(t , x)|R3 = 1, Leb ⊗ P− a.e.

〈u(t), ϕ〉 − 〈u0, ϕ〉 =

∫ t

0〈∇u, (∇ϕ)× u〉 ds

−∫ t

0〈∇u,∇(u × ϕ)× u〉 ds

+

∫ t

0〈G(u)Pu, ϕ〉ds +

∫ t

0〈G(u)h, ϕ〉 dW (s).

G(u)f = u × f + u × (u × f )

〈Pu,∇ϕ〉 = 〈u,∇ϕ〉


Notation

Given u ∈ H1 we define u ×∆u as a measurable functiontaking values in L2 such that

〈u ×∆u, ϕ〉 = 〈∇u,u × (∇ϕ)〉


Weak existence for d = 3

Theorem

Let u0 ∈ H1, h ∈ L∞ ∩W1,3 and |u0(x)| = 1. Then there exists a solution(Ω,F , (Ft )t≥0,P,W , u) to the LLG equation such that for all T > 0

E∫ T

0|u ×∆u|2 dt <∞,

u(t) = u0 +

∫ t

0u ×∆ uds −

∫ t

0u × (u ×∆u)ds

+

∫ t

0G(u)Pu ds +

∫ t

0G(u)h dW (s),

u ∈ Cα(

[0,T ],L2), α <

12.


Proof I

Uniform estimates for the Galerkin approximations un,

Tightness of the family of probability laws L (un) : n ≥ 1,

Identification of the limit


Proof II: Galerkin approximations

en∞n=1 eigenbasis of ∆N in L2 and

πn orthogonal projection onto Hn = lin e1, . . . ,en .

dun = (Gn (un) ∆un (un) + Gn (un) Pun) dt + Gn (un) h dW ,un(0) = πnu0

Gn(u)f = πn (un × f )− πn (un × (un × f ))

For every n ≥ 1 there exists a unique strong solution in Hn.


Proof IV: tightness

Lemma

For any p ≥ 2, q ∈ [2,6) and β > 12 the set of laws

L (un) : n ≥ 1 is tight on

Lp (0,T ; Lq) ∩ C(

0,T ; H−β)

.


Proof of tightness

For β > 12 , α < 1

2 and p > 2

supn

E |un|2Wα,p(0,T ;H−β) <∞.

Then for −β < γ < 1

Lp (0,T ; H1) ∩Wα,p (0,T ; H−β)⊂ Lp (0,T ; Hγ) ,

with compact embedding by Flandoli&Gatarek 1995 and tightness on

Lp (0,T ; Hγ) ⊂ Lp (0,T ; Lq)

follows. Again by Flandoli&Gatarek 1995

Wα,p (0,T ; H−β1)⊂ C

(0,T ; H−β

), β > β1, αp > 1,

with compact embedding.


Doss-Sussman method

Simplified stochastic Landau-Lifschitz-Gilbert equation:

du = [u ×∆u − u × (u ×∆u)]dt + (u × h) dW , t > 0, x ∈ D,

∂u∂n = 0, t ≥ 0, x ∈ ∂D,

u(0, x) = u0(x), x ∈ D.


Doss-Sussman method: auxiliary facts

Bx = x × a, x ∈ R3

Then etB is a group of isometries and

etB(x × y) =(

etBx)×(

etBy), x , y ∈ R3.

For h ∈ H2 putGφ = φ× h, φ ∈ L2

Then(etG) is again a group of isometries in L2 and

etGφ = φ+ (sint)Gφ+ (1− costt)G2φ


Doss-Sussman method III: transformation

Letv(t) = e−W (t)Gu(t).

Thendvdt

= v × R(t)v − v × (v × R(t)v) (2)

whereR(t)v = e−W (t)G∆eW (t)Gv


Doss-Sussman method: transformationcontinued.

Lemma

For φ ∈ H2

e−tG∆etGφ = ∆φ+

∫ t

0e−sGCesGφds,

with

Cφ = φ×∆h + 2∑

i

(∂φ

∂xi

)×(∂h∂xi

).

If |v |R3 = 1 then we obtaindvdt = R(t)v + v × R(t)v +

∣∣∇etBv∣∣2 v

v(0) = u0.(3)


Doss-Sussman: regularity

Theorem

Let h ∈ H2 and u0 ∈W1,4. Then for every ω there existsT = T (ω) > 0 such that equation (3) has a unique solution u on[0,T ) with the property

u ∈ C(

0,T ; W1,4)

and|v(t , x)|R3 = 1, t < T , x ∈ D.


Proof of Theorem 7

Equation (3) is a strongly elliptic quasi-linear systemShow that there exists a mild solution v ∈ C

(0,T ; W1,4)

Use maximal regularity and ultracontractivity of the heatsemigroup to "bootstrap" the regularity of solutions.Show that |v(t , x)| = 1.

Note that (2) can be written in the form

dvdt

= ∆v + v ×∆v + |∇v |2v + v × L(t , v) + v × (v × L(t , v))

with L linear and|L(t , v |L2 ≤ C|v |H1

where C is a finite random variable.


Theorem

The process u(t) = eW (t)Gv(t) is a unique solution of thestochastic Landau-Lifschitz-Gilbert equation on [0,T ) satisfyingfor every n ≥ 1 conditions

E∫ T∧n

0|∆Nv(s)|22 <∞

E supt≤T∧n

|∇v(t)|2 <∞,

Proof: takeu(t) = eW (t)Gv(t).

Use the Ito formula to obtain the estimates.


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