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The half-wave maps equation on $\mathbb{T}$: Global well-posedness in $H^{1/2}$ and almost periodicity

Patrick Gérard, Enno Lenzmann

Abstract

We consider the half-wave maps equation $$ \partial_t \mathbf{u} = \mathbf{u} \times |D| \mathbf{u} $$ for $\mathbf{u} : \mathbb{R} \times \mathbb{T} \to \mathbb{S}^2$, where $\mathbb{T}=\mathbb{R}/2 π\mathbb{Z}$ is the one-dimensional torus and $\mathbb{S}^2 \subset \mathbb{R}^3$ denotes the unit sphere. By extension from rational initial data, we construct a unique and continuous flow map for data in the critical energy space $H^{1/2}(\mathbb{T}; \mathbb{S}^2)$. Moreover, we show almost periodicity in time of these solutions. For the dense subset of rational initial data, we establish quasi-periodicity in time and a-priori bounds on $\| \mathbf{u}(t) \|_{H^s(\mathbb{T})}$ for any $s >0$. Our analysis relies crucially on an explicit formula arising from the Lax pair structure acting on a Hardy space of vector-valued holomorphic functions on the unit disk. As a central ingredient, we develop a general {\em stability principle} for explicit formulae associated with completely integrable PDEs possessing a Lax pair structure on Hardy spaces, including the Benjamin--Ono equation, Calogero--Sutherland DNLS, and the half-wave-maps equation posed on $\mathbb{T}$. Our results extend to the matrix-valued half-wave maps equation $$ \partial_t \mathbf{U} = -\frac{i}{2} [ \mathbf{U}, |D| \mathbf{U} ] $$ with target manifold given by the complex Grassmannians $\mathsf{Gr}_k(\mathbb{C}^d)$, thereby generalizing the special case $\mathbb{S}^2 \cong \mathbb{CP}^1 \cong \mathsf{Gr}_1(\mathbb{C}^2)$. In a companion work, we prove global well-posedness for the half-wave maps equation posed on $\mathbb{R}$ in the scaling-critical energy space $\dot{H}^{1/2}$, by establishing a stability principle for explicit formulae on Hardy spaces in the complex half-plane $\mathbb{C}_+$.

The half-wave maps equation on $\mathbb{T}$: Global well-posedness in $H^{1/2}$ and almost periodicity

Abstract

We consider the half-wave maps equation for , where is the one-dimensional torus and denotes the unit sphere. By extension from rational initial data, we construct a unique and continuous flow map for data in the critical energy space . Moreover, we show almost periodicity in time of these solutions. For the dense subset of rational initial data, we establish quasi-periodicity in time and a-priori bounds on for any . Our analysis relies crucially on an explicit formula arising from the Lax pair structure acting on a Hardy space of vector-valued holomorphic functions on the unit disk. As a central ingredient, we develop a general {\em stability principle} for explicit formulae associated with completely integrable PDEs possessing a Lax pair structure on Hardy spaces, including the Benjamin--Ono equation, Calogero--Sutherland DNLS, and the half-wave-maps equation posed on . Our results extend to the matrix-valued half-wave maps equation with target manifold given by the complex Grassmannians , thereby generalizing the special case . In a companion work, we prove global well-posedness for the half-wave maps equation posed on in the scaling-critical energy space , by establishing a stability principle for explicit formulae on Hardy spaces in the complex half-plane .
Paper Structure (36 sections, 35 theorems, 272 equations)

This paper contains 36 sections, 35 theorems, 272 equations.

Table of Contents

  1. Introduction
  2. Formulation of the problem
  3. Matrix-valued generalization of (HWM)
  4. Lax pair structure
  5. Main results
  6. Global well-posedness in $H^{1/2}$
  7. Long-time behavior of solutions
  8. Road map and remarks
  9. GWP for rational data
  10. GWP in $H^{1/2}$
  11. Almost periodicity
  12. On the zero-dispersion limit of (BO)
  13. Notation and preliminaries
  14. Vector-valued Hardy spaces and shift operators
  15. Toeplitz and Hankel operators
  16. ...and 21 more sections

Key Result

Lemma 2.1

Let $d \geq 2$ and $1 \leq k \leq d-1$ be integers. Then, for every $\mathbf{U}_0 \in \mathcal{R}at(\mathbb{T}; \mathsf{Gr}_k(\mathbb{C}^d))$, there exists a unique solution $\mathbf{U} \in C^\infty(\mathbb{R} \times \mathbb{T})$ of eq:HWM with initial datum $\mathbf{U}(0) =\mathbf{U}_0$. Moreover, and it holds that $\mathbf{U}(t) \in \mathcal{R}at(\mathbb{T}; \mathsf{Gr}_k(\mathbb{C}^d))$ for al

Theorems & Definitions (73)

  • Lemma 2.1: GWP for rational data
  • Theorem 2.1: GWP in $H^{1/2}$
  • Theorem 2.2: Lax evolution
  • Theorem 2.3: Almost periodicity
  • Theorem 2.4: Quasi-periodicity and a-priori bounds for rational data
  • Remark
  • Theorem : Stability principle for explicit formulae
  • Lemma 4.1
  • Remark
  • proof
  • ...and 63 more