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3D Stochastic Landau-Lifshitz-Gilbert Equations coupled with Maxwell's Equations with full energy

Zdzislaw Brzezniak, Ben Goldys, Liang Li

Abstract

We consider 3D stochastic Landau-Lifshitz-Gilbert equations coupled with the Maxwell equations with the full energy. We have proved the existence and some further regularities of the weak solution.

3D Stochastic Landau-Lifshitz-Gilbert Equations coupled with Maxwell's Equations with full energy

Abstract

We consider 3D stochastic Landau-Lifshitz-Gilbert equations coupled with the Maxwell equations with the full energy. We have proved the existence and some further regularities of the weak solution.

Paper Structure

This paper contains 14 sections, 51 theorems, 301 equations.

Table of Contents

  1. Introduction
  2. Statement of the problem and formulation of the main result
  3. Galerkin approximation
  4. A priori estimates
  5. Tightness results
  6. Construction of new probability space and processes
  7. The existence of a weak solution
  8. Step 1
  9. Step 2
  10. Step 3
  11. Some further regularities of the weak solution
  12. Proof of the main result
  13. Acknowledgments
  14. Appendix: Proofs of some auxiliary results

Key Result

Lemma 2.5

For $M\in \mathbb{V}$, if we define $\Delta M\in \mathbb{V}^*$ by Then the total energy $\mathcal{E}:\mathbb{V}\times\mathbb{L}^2(\mathbb{R}^3)\times\mathbb{L}^2(\mathbb{R}^3)\longrightarrow\mathbb{R}$ defined in eq_te has partial derivatives of 2nd order with respect to $M, B$ and $E$ well defined and: for $u,v\in \mathbb{V}$,

Theorems & Definitions (113)

  • Definition 2.3: Magnetic Induction
  • Definition 2.4
  • Lemma 2.5
  • Definition 2.6: Effective field
  • Remark 2.8
  • Definition 2.9: Weak martingale solution of equation \ref{['eq:SLLG']}
  • Theorem 2.10
  • Remark 2.11
  • Remark 2.12
  • Lemma 3.1
  • ...and 103 more