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On well-posedness of the s-Schrödinger maps in the subcritical regime

Ahmed Dughayshim

TL;DR

The paper proves local well-posedness for the s-Schrödinger map in dimensions n≥3 within the subcritical regime by reducing the geometric PDE to a scalar Schrödinger-type equation via stereographic projection. A sophisticated resolution-space framework is built (X_k, Y^e_{k,k'}, Z_k, F^σ, N^σ) to handle linear, smoothing, and maximal estimates, along with careful commutator and bilinear controls. The nonlinear terms, which feature derivatives, are bounded using a combination of dyadic, multilinear, and algebraic estimates, yielding a contraction in the appropriate function space for small Besov data u0. Consequently, the authors obtain local existence, uniqueness, and Lipschitz continuity of the solution map for initial data in B^σ_Q with σ≥(n+1)/2, and show how the stereographic reduction back to the original map recovers a regular solution u : R^{n+1}→S^2. This work advances the understanding of subcritical regimes for geometric Schrödinger-type maps and paves the way for refined well-posedness results using compensation-type structures.

Abstract

We study well-posedness of the $s$-Schrödinger map equation in dimension $n \geq 3$ in the subcritical regime, more precisely we establish a local well-posedness result when the initial data is $u_{0} \in B^σ_{2,1}$ with $ σ\geq \frac{n+1}{2}$ and $ \Vert u_{0} \Vert_{B^σ_{2,1}} \ll 1.$

On well-posedness of the s-Schrödinger maps in the subcritical regime

TL;DR

The paper proves local well-posedness for the s-Schrödinger map in dimensions n≥3 within the subcritical regime by reducing the geometric PDE to a scalar Schrödinger-type equation via stereographic projection. A sophisticated resolution-space framework is built (X_k, Y^e_{k,k'}, Z_k, F^σ, N^σ) to handle linear, smoothing, and maximal estimates, along with careful commutator and bilinear controls. The nonlinear terms, which feature derivatives, are bounded using a combination of dyadic, multilinear, and algebraic estimates, yielding a contraction in the appropriate function space for small Besov data u0. Consequently, the authors obtain local existence, uniqueness, and Lipschitz continuity of the solution map for initial data in B^σ_Q with σ≥(n+1)/2, and show how the stereographic reduction back to the original map recovers a regular solution u : R^{n+1}→S^2. This work advances the understanding of subcritical regimes for geometric Schrödinger-type maps and paves the way for refined well-posedness results using compensation-type structures.

Abstract

We study well-posedness of the -Schrödinger map equation in dimension in the subcritical regime, more precisely we establish a local well-posedness result when the initial data is with and

Paper Structure

This paper contains 14 sections, 40 theorems, 448 equations.

Table of Contents

  1. Introduction
  2. Reduction to a Schrödinger equation
  3. Preliminaries
  4. Definition of the spaces:
  5. Estimates between the resolution spaces
  6. Main estimates for the resolution spaces
  7. Local smoothing, maximal estimates
  8. Commutator estimates in resolution spaces
  9. The algebra properties of F^sigma spaces
  10. Bilinear estimate
  11. Dyadic estimates
  12. Nonlinear estimates
  13. Proof of Theorem \ref{['Redeq']} and \ref{['MainResult']}
  14. Product structures

Key Result

Theorem 1.1

Let $n \geq 3$, $s\in (1/2,1)$ then for any $\sigma_{0} \geq \frac{n+1}{2}$ there exists $\epsilon_{0} := \epsilon_{0}(n,s, \sigma_{0}) > 0$ so that the following holds

Theorems & Definitions (80)

  • Theorem 1.1
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • proof : Proof of Theorem \ref{['MainResult']}
  • Lemma 3.1
  • Remark 3.2
  • Lemma 3.3
  • proof
  • ...and 70 more