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)
Ben Goldys (UNSW, Sydney) Stochastic Landau-Lifshitz-Gilbert Equation
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
Ben Goldys (UNSW, Sydney) Stochastic Landau-Lifshitz-Gilbert Equation
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
Ben Goldys (UNSW, Sydney) Stochastic Landau-Lifshitz-Gilbert Equation
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.
Ben Goldys (UNSW, Sydney) Stochastic Landau-Lifshitz-Gilbert Equation
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.
Ben Goldys (UNSW, Sydney) Stochastic Landau-Lifshitz-Gilbert Equation
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.
Ben Goldys (UNSW, Sydney) Stochastic Landau-Lifshitz-Gilbert Equation
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
Ben Goldys (UNSW, Sydney) Stochastic Landau-Lifshitz-Gilbert Equation
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
Ben Goldys (UNSW, Sydney) Stochastic Landau-Lifshitz-Gilbert Equation
Thermal noise
E (φ) = · · · −∫
DH · φ
Néel 1946: H = noise.
H = hdW
h : D → R3, W Brownian Motion
Ben Goldys (UNSW, Sydney) Stochastic Landau-Lifshitz-Gilbert Equation
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
Ben Goldys (UNSW, Sydney) Stochastic Landau-Lifshitz-Gilbert Equation
Non-local term
∆v = ∇ · u, in Rd , u ∈ H1
Formally∇v = ∇∆u−1∇ · u
∇v = Pu, restricted to D
P =k|k |⊗ k|k |
Ben Goldys (UNSW, Sydney) Stochastic Landau-Lifshitz-Gilbert Equation
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)
Ben Goldys (UNSW, Sydney) Stochastic Landau-Lifshitz-Gilbert Equation
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 .
Ben Goldys (UNSW, Sydney) Stochastic Landau-Lifshitz-Gilbert Equation
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〉
Ben Goldys (UNSW, Sydney) Stochastic Landau-Lifshitz-Gilbert Equation
Notation
Given u ∈ H1 we define u ×∆u as a measurable functiontaking values in L2 such that
〈u ×∆u, ϕ〉 = 〈∇u,u × (∇ϕ)〉
Ben Goldys (UNSW, Sydney) Stochastic Landau-Lifshitz-Gilbert Equation
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.
Ben Goldys (UNSW, Sydney) Stochastic Landau-Lifshitz-Gilbert Equation
Proof I
Uniform estimates for the Galerkin approximations un,
Tightness of the family of probability laws L (un) : n ≥ 1,
Identification of the limit
Ben Goldys (UNSW, Sydney) Stochastic Landau-Lifshitz-Gilbert Equation
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.
Ben Goldys (UNSW, Sydney) Stochastic Landau-Lifshitz-Gilbert Equation
Proof III: uniform estimates
Lemma
Let h ∈ L∞ ∩W1,3 and u0 ∈ H1. Then for p ≥ 1, β > 12 and T > 0
|un(t)|L2 = |un(0)|L2 , P− a.s.
supn
E
[sup
t∈[0,T ]
|∇un(t)|2pL2
]<∞,
supn
E∫ T
0|un(t)×∆un(t)|L2 dt <∞,
supn
E
(∫ T
0|un(t)×
(un(t)×∆un(t)
)|2L3/2 dt
)p/2
<∞.
supn
E∫ T
0|πn(un(t)×
(un(t)×∆un(t)
))|2H−β dt <∞.
Ben Goldys (UNSW, Sydney) Stochastic Landau-Lifshitz-Gilbert Equation
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)
.
Ben Goldys (UNSW, Sydney) Stochastic Landau-Lifshitz-Gilbert Equation
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.
Ben Goldys (UNSW, Sydney) Stochastic Landau-Lifshitz-Gilbert Equation
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.
Ben Goldys (UNSW, Sydney) Stochastic Landau-Lifshitz-Gilbert Equation
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
Ben Goldys (UNSW, Sydney) Stochastic Landau-Lifshitz-Gilbert Equation
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
Ben Goldys (UNSW, Sydney) Stochastic Landau-Lifshitz-Gilbert Equation
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)
Ben Goldys (UNSW, Sydney) Stochastic Landau-Lifshitz-Gilbert Equation
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.
Ben Goldys (UNSW, Sydney) Stochastic Landau-Lifshitz-Gilbert Equation
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.
Ben Goldys (UNSW, Sydney) Stochastic Landau-Lifshitz-Gilbert Equation
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.
Ben Goldys (UNSW, Sydney) Stochastic Landau-Lifshitz-Gilbert Equation