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 Joint work with Prof. Beniamin FARAH El RAFEI School of Mathematics and Statistics, UNSW Sydney Stochastic Landau–Lifshitz Equation on Real Line 13 February 2020 1 / 22
Deterministic Landau-Lifshitz equation (LL) Magnetic domain: D ⊆ R d , d ≥ 1. Magnetisation: u : R + × D → R 3 . Landau-Lifshitz equation: d u ( t ) = λ 1 u ( t ) × H eff − λ 2 u ( t ) × ( u ( t ) × H eff ) dt where × is the cross product in R 3 , λ 1 , λ 2 > 0 and H eff is the effective field such that H eff = −∇ E total Joint work with Prof. Beniamin FARAH El RAFEI School of Mathematics and Statistics, UNSW Sydney Stochastic Landau–Lifshitz Equation on Real Line 13 February 2020 2 / 22
Deterministic Landau-Lifshitz equation (LL) ∂ 2 u For H eff = −∇ E exch = ∆ u , where ∆ u = � d . i =1 ∂ x 2 i Landau-Lifshitz equation: d u ( t ) = λ 1 u ( t ) × ∆ u ( t ) − λ 2 u ( t ) × ( u ( t ) × ∆ u ( t )) . dt Initial conditions: u (0 , x ) = u 0 ( x ) , | u 0 ( x ) | = 1 . Property: | u ( t , x ) | = 1 ∀ x ∈ D , ∀ t > 0 . Joint work with Prof. Beniamin FARAH El RAFEI School of Mathematics and Statistics, UNSW Sydney 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 Joint work with Prof. Beniamin FARAH El RAFEI School of Mathematics and Statistics, UNSW Sydney 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 Joint work with Prof. Beniamin FARAH El RAFEI School of Mathematics and Statistics, UNSW Sydney Stochastic Landau–Lifshitz Equation on Real Line 13 February 2020 5 / 22
Stochastic Landau-Lifshitz equation (SLL) ζ : white noise. H eff = ∆ u + ζ . Physical problems: λ 2 is small. Stochastic Landau-Lifshitz equation: d u ( t ) = λ 1 u ( t ) × (∆ u ( t ) + ζ ) − λ 2 u ( t ) × ( u ( t ) × (∆ u ( t ))) . dt Joint work with Prof. Beniamin FARAH El RAFEI School of Mathematics and Statistics, UNSW Sydney 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: d u ( t ) = ( λ 1 u ( t ) × ∆ u ( t ) − λ 2 u ( t ) × ( u ( t ) × ∆ u ( t )) dt + λ 1 u ( t ) × ◦ dW ( t ) with ∞ � W ( t ) = W i ( t ) g i i =1 where W i sequence of independent one dimensional Brownian motion defined on a common probability space and g i : D → R 3 are given functions such that the sequence g i ⊂ H 1 and � ∞ i =1 | g i | 2 H 1 < ∞ . Initial conditions: u (0 , x ) = u 0 ( x ) , | u 0 ( x ) | = 1 . Joint work with Prof. Beniamin FARAH El RAFEI School of Mathematics and Statistics, UNSW Sydney 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 → R 3 W : Ω × R + → R g : D → R 3 Stochastic Landau-Lifshitz equation: d u ( t ) = ( u ( t ) × ∆ u ( t ) − λ u ( t ) × ( u ( t ) × ∆ u ( t )) + 1 2(( u ( t ) × g ) × g )) dt + ( u ( t ) × g ) dW ( t ) Initial conditions: u (0 , x ) = u 0 ( x ) , | u 0 ( x ) | = 1 . Joint work with Prof. Beniamin FARAH El RAFEI School of Mathematics and Statistics, UNSW Sydney 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 Joint work with Prof. Beniamin FARAH El RAFEI School of Mathematics and Statistics, UNSW Sydney 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 { x i } i ∈ Z where x i = ih . We denote by Z h := { x i } i ∈ Z . Let u h : Ω × R + × Z h → R 3 and g : Z h → R 3 . We define D + u h ( x ) := u h ( x + h ) − u h ( x ) , h D − u h ( x ) := u h ( x ) − u h ( x − h ) , h ∆ u h ( x ) := D + D − u h ( x ) = D − D + u h ( x ) ˜ = u h ( x + h ) − 2 u h ( x ) + u h ( x − h ) . h 2 We define 1 2 | u h | L ∞ | u h ( x ) | , | u h | L 2 � | u h ( x ) | 2 h = sup h = h . x ∈ Z h x ∈ Z h Joint work with Prof. Beniamin FARAH El RAFEI School of Mathematics and Statistics, UNSW Sydney Stochastic Landau–Lifshitz Equation on Real Line 13 February 2020 10 / 22
Discretized problem We define v : Z h → R 3 : | v | E h < ∞ � � E h = E h = | D + v | 2 with | v | 2 h + | v | 2 h , and E h the space of E h − valued processes v L 2 L ∞ endowed with the norm | v | 2 E h = sup t ≤ T E | v ( t ) | 2 E h . We get d u h ( t , x i ) = ( u h ( t , x i ) × ˜ ∆ u h ( t , x i ) − λ u h ( t , x i ) × ( u h ( t , x i ) × ˜ ∆ u h ( t , x i )) + 1 2 (( u h ( t , x i ) × g ( x i )) × g ( x i ))) dt + ( u h ( t , x i ) × g ( x i )) dW ( t ) , u h (0 , x i ) = u 0 ( x i ) , | u 0 ( x i ) | = 1 . Result: This problem involves an SDE which has a unique strong global solution u h ( t ) , t > 0 on E h . Joint work with Prof. Beniamin FARAH El RAFEI School of Mathematics and Statistics, UNSW Sydney Stochastic Landau–Lifshitz Equation on Real Line 13 February 2020 11 / 22
Energy estimates Lemma Assume that g ∈ H 1 , T ∈ (0 , ∞ ) and | u 0 ( x ) | = 1 . For all x i ∈ Z h and every t ∈ [0 , T ] , we have | u h ( t , x i ) | = 1 . (0.1) Joint work with Prof. Beniamin FARAH El RAFEI School of Mathematics and Statistics, UNSW Sydney Stochastic Landau–Lifshitz Equation on Real Line 13 February 2020 12 / 22
Energy estimates Lemma Assume that g ∈ H 1 , T ∈ (0 , ∞ ) and | u 0 ( x ) | = 1 . For all x i ∈ Z h and every t ∈ [0 , T ] , we have | u h ( t , x i ) | = 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 � � | D + u h ( t ) | 2 p E sup ≤ C , (0.2) L 2 h t ∈ [0 , T ] �� � T � p � | ˜ ∆ u h ( t ) | 2 E h dt ≤ C . (0.3) L 2 0 Joint work with Prof. Beniamin FARAH El RAFEI School of Mathematics and Statistics, UNSW Sydney Stochastic Landau–Lifshitz Equation on Real Line 13 February 2020 12 / 22
Piecewise linear Interpolation We define r h the interpolation operator by r h u h ( t , x ) = u h ( t , x i ) , ∀ x ∈ [ x i ; x i +1 ) . Then, we get dr h u h ( t , x ) = ( r h u h ( t , x ) × r h ˜ ∆ u h ( t , x ) − λ r h u h ( t , x ) × ( r h u h ( t , x ) × r h ˜ ∆ u h ( t , x )) + 1 2 (( r h u h ( t , x ) × r h g ( x )) × r h g ( x ))) dt +( r h u h ( t , x ) × r h g ( x )) dW ( t ) , r h u h (0 , x ) = u 0 ( x ) , | u 0 ( x ) | = 1 . Joint work with Prof. Beniamin FARAH El RAFEI School of Mathematics and Statistics, UNSW Sydney Stochastic Landau–Lifshitz Equation on Real Line 13 February 2020 13 / 22
Convergence We define � L 2 m = { u : R → R 3 | | u ( x ) | 2 ρ m ( x ) dx < ∞} , R where ρ m ( x ) = e − | x | m , m > 0, and for any n � t � � | r h D + u h ( t ) | L 2 , | r h ˜ ∆ u h ( s ) | 2 τ h n = inf { t > 0 | max L 2 ds > n } 0 with h → 0 τ h τ n = lim a . s . n Lemma For any n sufficiently large, we have P ( τ n > 0) = 1 . Joint work with Prof. Beniamin FARAH El RAFEI School of Mathematics and Statistics, UNSW Sydney Stochastic Landau–Lifshitz Equation on Real Line 13 February 2020 14 / 22
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