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Local well-posedness of the Schrödinger flow into $\mathbb{S}^2$ with natural boundary conditions

Bo Chen, Youde Wang

Abstract

In this paper, we develop a new approximation scheme to resolve the local well-posedness problem for the Landau-Lifshitz equation (i.e., the Schrödinger flow into the standard unit 2-sphere $\mathbb{S}^2\subset \mathbb{R}^3$) with natural boundary conditions.

Local well-posedness of the Schrödinger flow into $\mathbb{S}^2$ with natural boundary conditions

Abstract

In this paper, we develop a new approximation scheme to resolve the local well-posedness problem for the Landau-Lifshitz equation (i.e., the Schrödinger flow into the standard unit 2-sphere ) with natural boundary conditions.
Paper Structure (33 sections, 26 theorems, 310 equations)

This paper contains 33 sections, 26 theorems, 310 equations.

Table of Contents

  1. Introduction
  2. Definitions and backgrounds
  3. Related works
  4. Main results
  5. Related models
  6. Strategies in the proof of Theorem \ref{['main-thm']}
  7. New approximation schemes for LL equation
  8. The outline of the proof of Theorem \ref{['main-thm']}
  9. Key challenges and innovative techniques
  10. Preliminary
  11. Geometric setup and Sobolev spaces
  12. Preliminary lemmas
  13. Sobolev-interpolation inequalities
  14. Compatibility conditions
  15. Compatibility conditions of the initial data
  16. ...and 18 more sections

Key Result

Theorem 1.1

Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^3$. Suppose that $u_0\in W^{5,2}(\Omega,\mathbb{S}^2)$, which satisfies the $1$-th order compatibility conditions, i.e. where $\nabla$ is the pull-back connection on ${u_0}^*(T\mathbb{S}^2)$. Then there exists a positive time $T_0$ depending only on $\Vert u_0\Vert_{W^{5,2}(\Omega)}$ such that the problem eq-LL admits a unique solution $u$ on

Theorems & Definitions (44)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Lemma 2.1
  • proof
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4: Theorem II.5.16 in BF
  • Lemma 3.5: Theorem II.5.14 in BF
  • ...and 34 more