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Global Solutions to the 3D Half-Wave Maps Equation with Angular Regularity

Katie Marsden

Abstract

The half-wave maps equation is a nonlocal geometric equation arising in the continuum dynamics of Haldane-Shashtry and Calogero-Moser spin systems. In high dimensions $n\geq4$, global wellposedness for data which is small in the critical Besov space $\dot{B}^{n/2}_{2,1}$ is known since the works of Krieger, Sire and Kiesenhofer [13,9]. There is a major obstruction in extending these results to three dimensions due to the loss of the crucial $L^2_tL^\infty_x$ Strichartz estimate. In this work, we make progress on this case by proving that the equation is "weakly" globally well-posed (in the sense of Tao [28]) for initial data which is not only small in $\dot{B}^{3/2}_{2,1}$ but also possesses some angular regularity and weighted decay of derivatives. We use Sterbenz's improved Strichartz estimates in conjunction with certain commuting vector fields to develop trilinear estimates in weighted Strichartz spaces which avoid the use of the $L^2_tL^\infty_x$ endpoint.

Global Solutions to the 3D Half-Wave Maps Equation with Angular Regularity

Abstract

The half-wave maps equation is a nonlocal geometric equation arising in the continuum dynamics of Haldane-Shashtry and Calogero-Moser spin systems. In high dimensions , global wellposedness for data which is small in the critical Besov space is known since the works of Krieger, Sire and Kiesenhofer [13,9]. There is a major obstruction in extending these results to three dimensions due to the loss of the crucial Strichartz estimate. In this work, we make progress on this case by proving that the equation is "weakly" globally well-posed (in the sense of Tao [28]) for initial data which is not only small in but also possesses some angular regularity and weighted decay of derivatives. We use Sterbenz's improved Strichartz estimates in conjunction with certain commuting vector fields to develop trilinear estimates in weighted Strichartz spaces which avoid the use of the endpoint.
Paper Structure (24 sections, 43 theorems, 367 equations, 1 figure)

This paper contains 24 sections, 43 theorems, 367 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation
  3. Preliminaries
  4. Angular Derivatives and Commuting Vector Fields
  5. Function Spaces and Linear Estimates
  6. Angular Multipliers
  7. Reduction to main proposition
  8. Discarding some error terms
  9. The half-wave maps contributions are negligible
  10. Showing that $P_0(HWM_1(\phi))=error$
  11. Showing that $P_0(HWM_2(\phi))=error$
  12. Normal Forms
  13. Low-high-high term
  14. Low-low-high error term
  15. The gauge transformation
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $\phi_0:\mathbb{R}^3\rightarrow \mathbb{S}^2$ be a smooth initial datum which is constant outside a compact set. There exists $0<\epsilon<1$ such that whenever the problem eqn1.1 admits a global smooth solution. Moreover for any $s$ sufficiently close to $3/2$ it holds for all $t\in\mathbb{R}$.

Figures (1)

  • Figure 1: The admissible Strichartz pairs $(p,q)$ in $n=3$. The dark gray region depicts the standard admissible pairs, and the light gray region the extended range of radially admissible pairs. The endpoint $L^2_tL^\infty_x$ is radially but not standard admissible.

Theorems & Definitions (73)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1: $n=3$ Strichartz estimates with angular regularity
  • proof
  • Lemma 2.2: Riesz estimate for angular Sobolev spaces (Theorem 3.5.3, spheres_book)
  • Lemma 2.3: Monotonicity of Angular Sobolev Spaces
  • Corollary 2.3.1
  • Theorem 2.4: Linear Estimate
  • Proposition 2.5
  • ...and 63 more