Wong–Zakai approximation for the stochastic Landau–Lifshitz–Gilbert equations

Zdzislaw Brzezniak; Utpal Manna; Debopriya Mukherjee

doi:10.1016/j.jde.2019.01.025

Wong–Zakai approximation for the stochastic Landau–Lifshitz–Gilbert equations

Zdzislaw Brzezniak, Utpal Manna, Debopriya Mukherjee

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

In this work we study stochastic Landau–Lifshitz–Gilbert equations (SLLGEs) in one dimension, with non-zero exchange energy only. Firstly, by introducing a suitable transformation, we convert the SLLGEs to a highly nonlinear time dependent partial differential equation with random coefficients, which is not fully parabolic. We then prove that there exists a pathwise unique solution to this equation and that this solution enjoys the maximal regularity property. Following regular approximation of the Brownian motion and using reverse transformation, we show existence of strong solution of SLLGEs taking values in a two-dimensional unit sphere S ² in R ³ . The construction of the solution and its corresponding convergence results are based on Wong–Zakai approximation.

Original language	English
Pages (from-to)	776-825
Number of pages	50
Journal	Journal of Differential Equations
Volume	267
Issue number	2
Early online date	5 Feb 2019
DOIs	https://doi.org/10.1016/j.jde.2019.01.025
Publication status	Published - 5 Jul 2019

Keywords

Ferromagnetism
Maximal regularity
Stochastic Landau–Lifshitz–Gilbert equations
Wong–Zakai approximation

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10.1016/j.jde.2019.01.025

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title = "Wong–Zakai approximation for the stochastic Landau–Lifshitz–Gilbert equations",

abstract = " In this work we study stochastic Landau–Lifshitz–Gilbert equations (SLLGEs) in one dimension, with non-zero exchange energy only. Firstly, by introducing a suitable transformation, we convert the SLLGEs to a highly nonlinear time dependent partial differential equation with random coefficients, which is not fully parabolic. We then prove that there exists a pathwise unique solution to this equation and that this solution enjoys the maximal regularity property. Following regular approximation of the Brownian motion and using reverse transformation, we show existence of strong solution of SLLGEs taking values in a two-dimensional unit sphere S 2 in R 3 . The construction of the solution and its corresponding convergence results are based on Wong–Zakai approximation. ",

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author = "Zdzislaw Brzezniak and Utpal Manna and Debopriya Mukherjee",

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AU - Brzezniak, Zdzislaw

AU - Manna, Utpal

AU - Mukherjee, Debopriya

PY - 2019/7/5

Y1 - 2019/7/5

N2 - In this work we study stochastic Landau–Lifshitz–Gilbert equations (SLLGEs) in one dimension, with non-zero exchange energy only. Firstly, by introducing a suitable transformation, we convert the SLLGEs to a highly nonlinear time dependent partial differential equation with random coefficients, which is not fully parabolic. We then prove that there exists a pathwise unique solution to this equation and that this solution enjoys the maximal regularity property. Following regular approximation of the Brownian motion and using reverse transformation, we show existence of strong solution of SLLGEs taking values in a two-dimensional unit sphere S 2 in R 3 . The construction of the solution and its corresponding convergence results are based on Wong–Zakai approximation.

AB - In this work we study stochastic Landau–Lifshitz–Gilbert equations (SLLGEs) in one dimension, with non-zero exchange energy only. Firstly, by introducing a suitable transformation, we convert the SLLGEs to a highly nonlinear time dependent partial differential equation with random coefficients, which is not fully parabolic. We then prove that there exists a pathwise unique solution to this equation and that this solution enjoys the maximal regularity property. Following regular approximation of the Brownian motion and using reverse transformation, we show existence of strong solution of SLLGEs taking values in a two-dimensional unit sphere S 2 in R 3 . The construction of the solution and its corresponding convergence results are based on Wong–Zakai approximation.

KW - Ferromagnetism

KW - Maximal regularity

KW - Stochastic Landau–Lifshitz–Gilbert equations

KW - Wong–Zakai approximation

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DO - 10.1016/j.jde.2019.01.025

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SN - 0022-0396

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EP - 825

JO - Journal of Differential Equations

JF - Journal of Differential Equations

IS - 2

ER -

Wong–Zakai approximation for the stochastic Landau–Lifshitz–Gilbert equations

Abstract

Keywords

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