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Smooth solutions to the Schrödinger flow for maps from smooth bounded domains in Euclidean spaces into $\mathbb{S}^2$

Bo Chen, Youde Wang

TL;DR

The paper establishes local well-posedness for very regular solutions to the initial-Neumann Schrödinger flow $\partial_t u = u\times\Delta u$ with targets in $\mathbb{S}^2$ for domains $\Omega\subset\mathbb{R}^m$ ($m=1,2,3$) under precise boundary compatibility in Sobolev spaces. It introduces a parabolic perturbation $\partial_t u_\varepsilon = \varepsilon\tau(u_\varepsilon) + J(u_\varepsilon)\tau(u_\varepsilon)$ to obtain uniform high-order energy estimates and proves that, as $\varepsilon\to 0$, these converge to a unique very regular solution of the Schrödinger flow; stronger compatibility yields higher regularity via induction. For the one-dimensional setting, the authors prove global existence of smooth solutions by exploiting a conserved energy law $\partial_t\int_I |\partial_t u|^2dx - \frac{1}{4}\int_I |\partial_x u|^4dx = 0$, along with preserved $\int_I |\partial_x u|^2dx$ and higher-order energy bounds, ensuring global well-posedness. The results extend known local well-posedness theories and provide a robust framework for high-regularity and boundary-compatible Schrödinger flows, with explicit constructions of compatibility conditions and uniform estimates. The work advances the mathematical understanding of geometric Schrödinger-type evolutions with Neumann boundaries and offers tools potentially applicable to related geometric flows and boundary-value problems.

Abstract

The results of this paper are twofold. One is that we show the local existence and uniqueness of very regular or smooth solution to the initial-Neumann boundary value problem of the Schrödinger flow for maps from a smooth bounded domain $Ω\subset \mathbb{R}^m$ with $m=1,2,3$ into $\mathbb{S}^2$ in the scale of Sobolev spaces. In this part, we provide a precise description of the compatibility conditions at the boundary for the initial data. The other is that we further prove that the locally smooth solution to the initial-Neumann boundary value problem of the 1-dimensional Schrödinger flow can be extended to a global smooth one.

Smooth solutions to the Schrödinger flow for maps from smooth bounded domains in Euclidean spaces into $\mathbb{S}^2$

TL;DR

The paper establishes local well-posedness for very regular solutions to the initial-Neumann Schrödinger flow with targets in for domains () under precise boundary compatibility in Sobolev spaces. It introduces a parabolic perturbation to obtain uniform high-order energy estimates and proves that, as , these converge to a unique very regular solution of the Schrödinger flow; stronger compatibility yields higher regularity via induction. For the one-dimensional setting, the authors prove global existence of smooth solutions by exploiting a conserved energy law , along with preserved and higher-order energy bounds, ensuring global well-posedness. The results extend known local well-posedness theories and provide a robust framework for high-regularity and boundary-compatible Schrödinger flows, with explicit constructions of compatibility conditions and uniform estimates. The work advances the mathematical understanding of geometric Schrödinger-type evolutions with Neumann boundaries and offers tools potentially applicable to related geometric flows and boundary-value problems.

Abstract

The results of this paper are twofold. One is that we show the local existence and uniqueness of very regular or smooth solution to the initial-Neumann boundary value problem of the Schrödinger flow for maps from a smooth bounded domain with into in the scale of Sobolev spaces. In this part, we provide a precise description of the compatibility conditions at the boundary for the initial data. The other is that we further prove that the locally smooth solution to the initial-Neumann boundary value problem of the 1-dimensional Schrödinger flow can be extended to a global smooth one.
Paper Structure (23 sections, 33 theorems, 313 equations)

This paper contains 23 sections, 33 theorems, 313 equations.

Table of Contents

  1. Introduction
  2. Definitions and Background
  3. Main results on local regular solutions to the Schrödinger flow
  4. Global smooth solutions to 1-dimensional Schrödinger flow
  5. Preliminary
  6. Some preliminary lemmas
  7. Compatibility conditions
  8. Compatibility conditions of the initial data
  9. Uniform Compatibility conditions of the initial data
  10. Another compatibility conditions
  11. $H^5$-regular local solution
  12. Uniform $H^2$-estimates
  13. Uniform $H^3$-estimates
  14. Very regular local solution
  15. Estimates of equivalent norms
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^3$. Suppose that $u_0\in H^5(\Omega,\mathbb{S}^2)$, which satisfies the $1$-order compatibility condition defined in com-cond2, i.e. $\frac{\partial u_0}{\partial \nu}|_{\partial \Omega}=0$ and $\tilde{\nabla}_\nu \tau(u_0)|_{\partial\Omega}=0$, for $i=0,1,2$.

Theorems & Definitions (55)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Lemma 2.1
  • Lemma 2.2
  • Corollary 2.3
  • proof
  • ...and 45 more