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Stochastic Landau–Lifshitz Equation on Real Line

FARAH El RAFEI

School of Mathematics and Statistics, UNSW Sydney

Joint work with

• Prof. Beniamin Goldys

School of Mathematics and Statistics, The University of Sydney

• Prof. Thanh Tran

School of Mathematics and Statistics, UNSW Sydney

13 February 2020

FARAH El RAFEI School of Mathematics and Statistics, UNSW Sydney Joint work with Prof. Beniamin Stochastic Landau–Lifshitz Equation on Real Line 13 February 2020 1 / 22

Deterministic Landau-Lifshitz equation (LL)

Magnetic domain: D ⊆ Rd, d ≥ 1. Magnetisation: u : R+ × D → R3. Landau-Lifshitz equation: du(t) dt = λ1u(t) × Heff − λ2u(t) × (u(t) × Heff ) where × is the cross product in R3, λ1, λ2 > 0 and Heff is the effective field such that Heff = −∇Etotal

FARAH El RAFEI School of Mathematics and Statistics, UNSW Sydney Joint work with Prof. Beniamin Stochastic Landau–Lifshitz Equation on Real Line 13 February 2020 2 / 22

Deterministic Landau-Lifshitz equation (LL)

For Heff = −∇Eexch = ∆u, where ∆u = d

i=1

∂2u ∂x2

i

. Landau-Lifshitz equation: du(t) dt = λ1u(t) × ∆u(t) − λ2u(t) × (u(t) × ∆u(t)). Initial conditions: u(0, x) = u0(x), |u0(x)| = 1. Property: |u(t, x)| = 1 ∀x ∈ D, ∀t > 0.

FARAH El RAFEI School of Mathematics and Statistics, UNSW Sydney Joint work with Prof. Beniamin Stochastic Landau–Lifshitz Equation on Real Line 13 February 2020 3 / 22

Previous work

Bounded domain:

• A. Visintin. On Landau-Lifshitz’ equations for ferromagnetism.

Japan J. Appl. Math., 2(1):69–84, 1985

• F. Alouges and A. Soyeur. On global weak solutions for

Landau-Lifshitz equations: existence and nonuniqueness. Nonlinear Anal., 18(11):1071–1084, 1992

FARAH El RAFEI School of Mathematics and Statistics, UNSW Sydney Joint work with Prof. Beniamin Stochastic Landau–Lifshitz Equation on Real Line 13 February 2020 4 / 22

Previous work

Unbounded domain:

• F. Alouges and A. Soyeur. On global weak solutions for

Landau-Lifshitz equations: existence and nonuniqueness. Nonlinear Anal., 18(11):1071–1084, 1992

• A. Fuwa and M. Tsutsumi. Local well posedness of the Cauchy

problem for the Landau-Lifshitz equations. Differential Integral Equations, 18(4):379–404, 2005

FARAH El RAFEI School of Mathematics and Statistics, UNSW Sydney Joint work with Prof. Beniamin Stochastic Landau–Lifshitz Equation on Real Line 13 February 2020 5 / 22

Stochastic Landau-Lifshitz equation (SLL)

ζ: white noise. Heff = ∆u + ζ. Physical problems: λ2 is small. Stochastic Landau-Lifshitz equation: du(t) dt = λ1u(t) × (∆u(t) + ζ) − λ2u(t) × (u(t) × (∆u(t))).

FARAH El RAFEI School of Mathematics and Statistics, UNSW Sydney Joint work with Prof. Beniamin Stochastic Landau–Lifshitz Equation on Real Line 13 February 2020 6 / 22

Stochastic Landau-Lifshitz equation (SLL)

ζ = ˙ W with W Wiener process. Stochastic Landau-Lifshitz equation: du(t) = (λ1u(t) × ∆u(t) − λ2u(t) × (u(t) × ∆u(t))dt + λ1u(t) × ◦dW (t) with W (t) =

∞

• i=1

Wi(t)gi where Wi sequence of independent one dimensional Brownian motion defined on a common probability space and gi : D → R3 are given functions such that the sequence gi ⊂ H1 and ∞

i=1 |gi|2 H1 < ∞.

Initial conditions: u(0, x) = u0(x), |u0(x)| = 1.

FARAH El RAFEI School of Mathematics and Statistics, UNSW Sydney Joint work with Prof. Beniamin Stochastic Landau–Lifshitz Equation on Real Line 13 February 2020 7 / 22

Stochastic Landau-Lifshitz equation (SLL)

(Ω, F, P) a filtration probability space u : Ω × R+ × D → R3 W : Ω × R+ → R g : D → R3 Stochastic Landau-Lifshitz equation: du(t) = (u(t) × ∆u(t) − λu(t) × (u(t) × ∆u(t)) + 1 2((u(t) × g) × g))dt + (u(t) × g)dW (t) Initial conditions: u(0, x) = u0(x), |u0(x)| = 1.

FARAH El RAFEI School of Mathematics and Statistics, UNSW Sydney Joint work with Prof. Beniamin Stochastic Landau–Lifshitz Equation on Real Line 13 February 2020 8 / 22

Previous work

Bounded domain:

• Z. Brzeźniak, B. Goldys, and T. Jegaraj. Weak solutions of a

stochastic Landau-Lifshitz-Gilbert equation.

• Appl. Math. Res. Express. AMRX, (1):1–33, 2013
• Z. a. Brzeźniak, B. Goldys, and T. Jegaraj. Large deviations and

transitions between equilibria for stochastic Landau-Lifshitz-Gilbert equation.

• Arch. Ration. Mech. Anal., 226(2):497–558, 2017
• B. Goldys, K.-N. Le, and T. Tran. A finite element approximation for

the stochastic Landau-Lifshitz-Gilbert equation.

• J. Differential Equations, 260(2):937–970, 2016
• F. Alouges, A. de Bouard, and A. Hocquet. A semi-discrete scheme

for the stochastic Landau-Lifshitz equation.

• Stoch. Partial Differ. Equ. Anal. Comput., 2(3):281–315, 2014

FARAH El RAFEI School of Mathematics and Statistics, UNSW Sydney Joint work with Prof. Beniamin Stochastic Landau–Lifshitz Equation on Real Line 13 February 2020 9 / 22

Difference method

We work on SLL with D = R. Given h > 0, we consider {xi}i∈Z where xi = ih. We denote by Zh := {xi}i∈Z. Let uh : Ω × R+ × Zh → R3 and g : Zh → R3. We define D+uh(x) := uh(x + h) − uh(x) h , D−uh(x) := uh(x) − uh(x − h) h , ˜ ∆uh(x) := D+D−uh(x) = D−D+uh(x) = uh(x + h) − 2uh(x) + uh(x − h) h2 . We define |uh|L∞

h = sup

x∈Zh

|uh(x)|, |uh|L2

h =

 h

• x∈Zh

|uh(x)|2

 

1 2

.

FARAH El RAFEI School of Mathematics and Statistics, UNSW Sydney Joint work with Prof. Beniamin Stochastic Landau–Lifshitz Equation on Real Line 13 February 2020 10 / 22

Discretized problem

We define Eh =

• v : Zh → R3 : |v|Eh < ∞
• with |v|2

Eh = |D+v|2 L2

h + |v|2

L∞

h , and Eh the space of Eh−valued processes v

endowed with the norm |v|2

Eh = supt≤T E|v(t)|2 Eh .

We get

            

duh(t, xi) = (uh(t, xi) × ˜ ∆uh(t, xi) − λuh(t, xi) × (uh(t, xi) × ˜ ∆uh(t, xi)) +1

2((uh(t, xi) × g(xi)) × g(xi)))dt + (uh(t, xi) × g(xi))dW (t),

uh(0, xi) = u0(xi), |u0(xi)| = 1. Result: This problem involves an SDE which has a unique strong global solution uh(t), t > 0 on Eh.

FARAH El RAFEI School of Mathematics and Statistics, UNSW Sydney Joint work with Prof. Beniamin Stochastic Landau–Lifshitz Equation on Real Line 13 February 2020 11 / 22

Energy estimates

Lemma Assume that g ∈ H1, T ∈ (0, ∞) and |u0(x)| = 1. For all xi ∈ Zh and every t ∈ [0, T], we have |uh(t, xi)| = 1. (0.1)

FARAH El RAFEI School of Mathematics and Statistics, UNSW Sydney Joint work with Prof. Beniamin Stochastic Landau–Lifshitz Equation on Real Line 13 February 2020 12 / 22

Energy estimates

Lemma Assume that g ∈ H1, T ∈ (0, ∞) and |u0(x)| = 1. For all xi ∈ Zh and every t ∈ [0, T], we have |uh(t, xi)| = 1. (0.1) Moreover, given 1 ≤ p < ∞, there exists a constant C which does not depend on h but which may depend on g and T such that E

• sup

t∈[0,T]

|D+uh(t)|2p

L2

h

• ≤ C,

(0.2) E

T

| ˜ ∆uh(t)|2

L2

hdt

p

≤ C. (0.3)

FARAH El RAFEI School of Mathematics and Statistics, UNSW Sydney Joint work with Prof. Beniamin Stochastic Landau–Lifshitz Equation on Real Line 13 February 2020 12 / 22

Piecewise linear Interpolation

We define rh the interpolation operator by rhuh(t, x) = uh(t, xi), ∀x ∈ [xi; xi+1). Then, we get

                      

drhuh(t, x) = (rhuh(t, x) × rh ˜ ∆uh(t, x) −λrhuh(t, x) × (rhuh(t, x) × rh ˜ ∆uh(t, x)) +1

2((rhuh(t, x) × rhg(x)) × rhg(x)))dt

+(rhuh(t, x) × rhg(x))dW (t), rhuh(0, x) = u0(x), |u0(x)| = 1.

FARAH El RAFEI School of Mathematics and Statistics, UNSW Sydney Joint work with Prof. Beniamin Stochastic Landau–Lifshitz Equation on Real Line 13 February 2020 13 / 22

Convergence

We define L2

m = {u : R → R3|

• R

|u(x)|2ρm(x)dx < ∞}, where ρm(x) = e− |x|

m , m > 0, and for any n

τ h

n = inf{t > 0|

max

• |rhD+uh(t)|L2,

t

|rh ˜ ∆uh(s)|2

L2ds

• > n}

with τn = lim

h→0 τ h n

a.s. Lemma For any n sufficiently large, we have P(τn > 0) = 1.

FARAH El RAFEI School of Mathematics and Statistics, UNSW Sydney Joint work with Prof. Beniamin Stochastic Landau–Lifshitz Equation on Real Line 13 February 2020 14 / 22

Convergence

Lemma Assume g ∈ H1. For any n sufficiently large, there exists un ∈ L2(Ω, L∞((0, T ∧ τn

2 ), L2 m(R))) such that as h → 0, the following

limits exist E

• sup

t∈[0,T∧ τn

2 ]

|rhuh − un|2

L2

m

• → 0.

(0.4) E

• sup

t∈[0,T∧ τn

2 ]

|rhD+uh − ∇un|2

L2

• → 0.

(0.5) E

T∧ τn

2

|rh ˜ ∆uh − ∆un|2

L2

• → 0.

(0.6)

FARAH El RAFEI School of Mathematics and Statistics, UNSW Sydney Joint work with Prof. Beniamin Stochastic Landau–Lifshitz Equation on Real Line 13 February 2020 15 / 22

Existence of local strong solution

Lemma Assume u0(x) ∈ S2 for all x ∈ R, where S2 is the unit sphere in R3, ∇u0 ∈ L2(R) and g ∈ H1(R). For T > 0, there exists a solution un ∈ L2(Ω, L∞((0, T ∧ τn

2 ), L2 m(R))) to SLL problem such that

1

|un(t, x)| = 1 ,

FARAH El RAFEI School of Mathematics and Statistics, UNSW Sydney Joint work with Prof. Beniamin Stochastic Landau–Lifshitz Equation on Real Line 13 February 2020 16 / 22

Existence of local strong solution

Lemma Assume u0(x) ∈ S2 for all x ∈ R, where S2 is the unit sphere in R3, ∇u0 ∈ L2(R) and g ∈ H1(R). For T > 0, there exists a solution un ∈ L2(Ω, L∞((0, T ∧ τn

2 ), L2 m(R))) to SLL problem such that

1

|un(t, x)| = 1 ,

2

un ∈ C((0, T), L2

m) ,

P − a.s.

FARAH El RAFEI School of Mathematics and Statistics, UNSW Sydney Joint work with Prof. Beniamin Stochastic Landau–Lifshitz Equation on Real Line 13 February 2020 16 / 22

Existence of local strong solution

Lemma Assume u0(x) ∈ S2 for all x ∈ R, where S2 is the unit sphere in R3, ∇u0 ∈ L2(R) and g ∈ H1(R). For T > 0, there exists a solution un ∈ L2(Ω, L∞((0, T ∧ τn

2 ), L2 m(R))) to SLL problem such that

1

|un(t, x)| = 1 ,

2

un ∈ C((0, T), L2

m) ,

P − a.s.

3 for every p > 0

E

• sup

t∈[0,T∧ τn

2 ]

|∇un(t)|p

L2

• < ∞ ,

FARAH El RAFEI School of Mathematics and Statistics, UNSW Sydney Joint work with Prof. Beniamin Stochastic Landau–Lifshitz Equation on Real Line 13 February 2020 16 / 22

Existence of local strong solution

Lemma

4 for every p > 0

E

T∧ τn

2

|∆un(t)|2

L2dt

p

< ∞ ,

FARAH El RAFEI School of Mathematics and Statistics, UNSW Sydney Joint work with Prof. Beniamin Stochastic Landau–Lifshitz Equation on Real Line 13 February 2020 17 / 22

Existence of local strong solution

Lemma

4 for every p > 0

E

T∧ τn

2

|∆un(t)|2

L2dt

p

< ∞ ,

5 the following equation holds in L∞((0, T ∧ τn

2 ), L2 m) P a.s., for all

t ∈ [0, T ∧ τn

2 ]:

un(t) = un

0 +

t

un(s) × ∆un(s)ds − λ

t

un(s) × (un(s) × ∆un(s))ds + 1 2

t

(un(s) × g) × gds +

t

un(s) × gdW (s).

FARAH El RAFEI School of Mathematics and Statistics, UNSW Sydney Joint work with Prof. Beniamin Stochastic Landau–Lifshitz Equation on Real Line 13 February 2020 17 / 22

Uniqueness of solution

Theorem Let u1 and u2 be two solutions for SLL with the same initial values. Then, we have that u1 = u2 a.s.

FARAH El RAFEI School of Mathematics and Statistics, UNSW Sydney Joint work with Prof. Beniamin Stochastic Landau–Lifshitz Equation on Real Line 13 February 2020 18 / 22

Existence of global strong solution

Finally, we can take the limit when n → ∞ and deduce that we have the following theorem Theorem Assume u0(x) ∈ S2 for all x ∈ R, where S2 is the unit sphere in R3, ∇u0 ∈ L2(R) and g ∈ H1(R). For T > 0, there exists a solution u ∈ L2(Ω, L∞((0, T), L2

m(R))) to SLL problem such that

1

|u(t, x)| = 1 ,

2

u ∈ C((0, T), L2

m) ,

P − a.s.

3 for every p > 0

E

• sup

t∈[0,T]

|∇u(t)|p

L2

• < ∞ ,

FARAH El RAFEI School of Mathematics and Statistics, UNSW Sydney Joint work with Prof. Beniamin Stochastic Landau–Lifshitz Equation on Real Line 13 February 2020 19 / 22

Existence of global strong solution

Theorem

4 for every p > 0

E

T

|∆u(t)|2

L2dt

p

< ∞ ,

5 the following equation holds in L∞((0, T), L2

m) P a.s., for all

t ∈ [0, T]: u(t) = u0 +

t

u(s) × ∆u(s)ds − λ

t

u(s) × (u(s) × ∆u(s))ds + 1 2

t

(u(s) × g) × gds +

t

u(s) × gdW (s).

FARAH El RAFEI School of Mathematics and Statistics, UNSW Sydney Joint work with Prof. Beniamin Stochastic Landau–Lifshitz Equation on Real Line 13 February 2020 20 / 22

Continuous dependence on initial conditions

Theorem Let u0i : R → S2 (i = 1, 2) be such that u01 − u02 ∈ L2(R). Let u1 and u2 be two solutions for SLL with initial values u01 and u02 respectively. Then, u1 − u2 ∈ L2(R) and E

• sup

t∈[0,T]

|u1 − u2|2

L2

• ≤ C|u01 − u02|2

L2.

FARAH El RAFEI School of Mathematics and Statistics, UNSW Sydney Joint work with Prof. Beniamin Stochastic Landau–Lifshitz Equation on Real Line 13 February 2020 21 / 22

T HANK YOU

FARAH El RAFEI School of Mathematics and Statistics, UNSW Sydney Joint work with Prof. Beniamin Stochastic Landau–Lifshitz Equation on Real Line 13 February 2020 22 / 22