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A Study on the Well-Posedness of 1D Energy-Critical Half-Wave Maps Equations

Gaspard Ohlmann

Abstract

In this article, we study the well-posedness of the energy-critical half-wave maps equation (HWM) in dimension $1$. The half-wave maps equation emerges from the continuum limit of the Haldane Shastry spin chains and has been shown to arise as the continuum limit of Calogero-Moser classical spin systems. In higher dimension $d\geq 5$, it has been shown that (HWM) is well-posed by Krieger and Sire. This result has been improved by Krieger and Kiesenhofer to $d = 4$ but the Strichartz estimate on which the argument is built no longer holds for smaller dimensions. A Lax-pair structure has been revealed for this equation by Lenzmann and Gérard, indicating complete integrability and the fact that rational solutions stay rational for all time. The well-posedness of the (HWM) equation in lower dimensions remains an open problem. Here, we show the well-posedness of (HWM) in the rational case for finite times with separated poles, and for large and infinite times with distinct speeds of propagation.

A Study on the Well-Posedness of 1D Energy-Critical Half-Wave Maps Equations

Abstract

In this article, we study the well-posedness of the energy-critical half-wave maps equation (HWM) in dimension . The half-wave maps equation emerges from the continuum limit of the Haldane Shastry spin chains and has been shown to arise as the continuum limit of Calogero-Moser classical spin systems. In higher dimension , it has been shown that (HWM) is well-posed by Krieger and Sire. This result has been improved by Krieger and Kiesenhofer to but the Strichartz estimate on which the argument is built no longer holds for smaller dimensions. A Lax-pair structure has been revealed for this equation by Lenzmann and Gérard, indicating complete integrability and the fact that rational solutions stay rational for all time. The well-posedness of the (HWM) equation in lower dimensions remains an open problem. Here, we show the well-posedness of (HWM) in the rational case for finite times with separated poles, and for large and infinite times with distinct speeds of propagation.
Paper Structure (8 sections, 16 theorems, 176 equations)

This paper contains 8 sections, 16 theorems, 176 equations.

Table of Contents

  1. Two solitons case, infinite time
  2. No infinite time blow-up with different speeds, specific case
  3. Absence of finite time blow-up for N solitons with separated poles
  4. Case where only one pole is going toward the real axis, proof with an analytic expression for the energy norm
  5. Case where several poles are going toward the real axis: proof with an algebraic expression of the energy norm
  6. Uniform control for N solitons with separated poles
  7. Separated-poles case
  8. Well posedness at $t=+\infty$, general case

Key Result

Theorem 1

Let $m$ be a solution of HWM of the form expressionSol.

Theorems & Definitions (32)

  • Theorem 1
  • Theorem 1.1
  • proof
  • Theorem 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • proof
  • Remark 2.4
  • ...and 22 more