Stochastic Landau-Lifshitz-Gilbert
Equation
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 Brzeźniak (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 = H 1 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 filling
a domain D ⊂ Rd , d ≤ 3,
u(x) the magnetic moment at x ∈ D,
For temperatures not too high
|u(x)| = 1,
Ben Goldys (UNSW, Sydney)
x ∈D
Stochastic Landau-Lifshitz-Gilbert Equation
Energy functional I
Every configuration φ : D → R3 , φ ∈ H1 of magnetic moments
minimizes the energy functional
Z
Z
Z
a1
1
2
2
E (φ) =
|∇φ| dx +
|∇v | dx −
H · φdx
2 D
2 Rd
D
∆v (x) = ∇ · φ̄(x), x ∈ Rd
φ(x) if x ∈ D,
φ̄(x) =
0
if x ∈
/ D.
|φ(x)| = 1,
Ben Goldys (UNSW, Sydney)
x ∈ D.
Stochastic Landau-Lifshitz-Gilbert Equation
Energy functional II
Landau-Lifschitz 1935, Gilbert 1955
Z
a1
|∇φ|2 dx, exchange energy,
2 D
Z
1
|∇v |2 dx, magnetostatic energy,
2 Rd
H- given external field.
Ben Goldys (UNSW, Sydney)
Stochastic Landau-Lifshitz-Gilbert Equation
Landau-Lifschitz-Gilbert equation
H (u) = −Du E (u) = a1 ∆u − ∇v + H











∂u
∂t
= λ1 u × H (u) − λ2 u × (u × H (u)) on D
∂u
∂n
=0
on ∂D
|u0 (x)| = 1
on D
λ2 > 0
and from now on
λ1 = λ2 = 1.
Ben Goldys (UNSW, Sydney)
Stochastic Landau-Lifshitz-Gilbert Equation
Connection with harmonic maps problem
1
E (φ) =
2
Z
|∇φ|2 dx
D
∂u
= −u × (u × ∆u)
∂t
but
u × (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
= ∆u + |∇u|2 u
∂t
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, Harpes
2004: 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, large
deviations
A. Desimone, R. V. Kohn, S. Müller, F. Otto 2002, thin film
approximations
R. Moser 2004, thin film approximations, magnetic vortices
Ben Goldys (UNSW, Sydney)
Stochastic Landau-Lifshitz-Gilbert Equation
Thermal noise
E (φ) = · · · −
Z
H ·φ
D
Néel 1946: H = noise.
H = hdW
h : D → R3 ,
W
Ben Goldys (UNSW, Sydney)
Brownian Motion
Stochastic Landau-Lifshitz-Gilbert Equation
Stochastic Landau-Lifschitz-Gilbert-Equation I
H (u) = −Du E (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 =
1
DF (u) · F (u)dt + FdW
2
Ben Goldys (UNSW, Sydney)
Stochastic Landau-Lifshitz-Gilbert Equation
Non-local term
∆v = ∇ · ū,
in Rd ,
u ∈ H1
Formally
∇v = ∇∆u −1 ∇ · u
∇v = P ū,
restricted to D
b= k ⊗ k
P
|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
Ben Goldys (UNSW, Sydney)
Stochastic Landau-Lifshitz-Gilbert Equation
(1)
Integration by parts
∆N Neumann Laplacian
∂u
2
= 0, on ∂D
D (∆N ) = u ∈ H :
∂n
.
Lemma
If v ∈ H1 and u ∈ D (∆N ) then
Z
Z
hu × ∆N , v i dx = h∇u, (∇v ) × ui dx.
D
D
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 and
φ ∈ C ∞ D̄, R3
u(·) ∈ C [0, T ]; H−1,2 , P − a.s.
E sup |∇u(t)|2L2 < ∞,
t≤T
|u(t, x)|R3 = 1, Leb ⊗ P − a.e.
Z t
hu(t), ϕi − hu0 , ϕi =
h∇u, (∇ϕ) × ui ds
Z
0
t
−
h∇u, ∇(u × ϕ) × ui ds
0
t
Z
t
Z
hG(u)Pu, ϕids +
+
0
hG(u)h, ϕi ◦ dW (s).
0
G(u)f = u × f + u × (u × f )
hPu, ∇ϕi = hu, ∇ϕi
Ben Goldys (UNSW, Sydney)
Stochastic Landau-Lifshitz-Gilbert Equation
Notation
Given u ∈ H1 we define u × ∆u as a measurable function
taking values in L2 such that
hu × ∆u, ϕi = h∇u, u × (∇ϕ)i
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
Z T
E
|u × ∆u|2 dt < ∞,
0
t
Z
0
t
Z
0
t
G(u)h ◦ dW (s),
0
u ∈ C α [0, T ], L2 ,
Ben Goldys (UNSW, Sydney)
u × (u × ∆u)ds
0
Z
G(u)Pu ds +
+
t
Z
u × ∆ uds −
u(t) = u0 +
α<
1
.
2
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
2
{en }∞
n=1 eigenbasis of ∆N in L and
πn
orthogonal projection onto Hn = lin {e1 , . . . , en } .
dun
= (Gn (un ) ∆un (un ) + Gn (un ) Pun ) dt + Gn (un ) h ◦ dW ,
un (0) = πn u0
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 >
|un (t)|L2 = |un (0)|L2 ,
"
sup E
n
sup E
Z
sup E
n
0
Z
sup E
n
0
and T > 0
P − a.s.
#
sup |∇un (t)|2p
< ∞,
L2
t∈[0,T ]
Z
n
1
2
0
T
|un (t) × ∆un (t)|L2 dt < ∞,
!p/2
T
|un (t) × un (t) × ∆un (t) |2L3/2 dt
< ∞.
T
|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 >
{L (un ) : n ≥ 1} is tight on
1
2
the set of laws
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
2
sup E |un |W α,p (0,T ;H−β ) < ∞.
n
Then for −β < γ < 1
Lp 0, T ; H1 ∩ W α,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
W α,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
t ≥ 0, x ∈ ∂D,
∂n = 0,





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 ) = etB x × etB y ,
For h ∈ H2 put
Gφ = φ × h,
x, y ∈ R3 .
φ ∈ L2
Then etG is again a group of isometries in L2 and
etG φ = φ + (sint)Gφ + (1 − costt)G2 φ
Ben Goldys (UNSW, Sydney)
Stochastic Landau-Lifshitz-Gilbert Equation
Doss-Sussman method III: transformation
Let
v (t) = e−W (t)G u(t).
Then
dv
= v × R(t)v − v × (v × R(t)v )
dt
where
R(t)v = e−W (t)G ∆eW (t)G v
Ben Goldys (UNSW, Sydney)
Stochastic Landau-Lifshitz-Gilbert Equation
(2)
Doss-Sussman method: transformation
continued.
Lemma
For φ ∈ H2
e
−tG
Z
tG
∆e φ = ∆φ +
t
e−sG CesG φ ds,
0
with
Cφ = φ × ∆h + 2
X ∂φ i
∂xi
×
∂h
∂xi
.
If |v |R3 = 1 then we obtain
(
tB 2
dv
dt = R(t)v + v × R(t)v + ∇e v v
v (0) = u0 .
Ben Goldys (UNSW, Sydney)
Stochastic Landau-Lifshitz-Gilbert Equation
(3)
Doss-Sussman: regularity
Theorem
Let h ∈ H2 and u0 ∈ W1,4 . Then for every ω there exists
T = 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,
Ben Goldys (UNSW, Sydney)
t < T , x ∈ D.
Stochastic Landau-Lifshitz-Gilbert Equation
Proof of Theorem 7
Equation (3) is a strongly elliptic quasi-linear system
Show that there exists a mild solution v ∈ C 0, T ; W1,4
Use maximal regularity and ultracontractivity of the heat
semigroup to "bootstrap" the regularity of solutions.
Show that |v (t, x)| = 1.
Note that (2) can be written in the form
dv
= ∆v + v × ∆v + |∇v |2 v + v × L(t, v ) + v × (v × L(t, v ))
dt
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)G v (t) is a unique solution of the
stochastic Landau-Lifschitz-Gilbert equation on [0, T ) satisfying
for every n ≥ 1 conditions
Z
T ∧n
E
0
|∆N v (s)|22 < ∞
E sup |∇v (t)|2 < ∞,
t≤T ∧n
Proof: take
u(t) = eW (t)G v (t).
Use the Ito formula to obtain the estimates.
Ben Goldys (UNSW, Sydney)
Stochastic Landau-Lifshitz-Gilbert Equation