Global Well-Posedness and Soliton Resolution for the Half-Wave Maps Equation with Rational Data

TL;DR

This paper proves global well-posedness and soliton resolution for the energy-critical half-wave maps equation with rational initial data, extending results to matrix-valued equations targeting complex Grassmannians.

Contribution

It establishes the first global well-posedness and soliton resolution results for rational data in the half-wave maps equation, including generalized target spaces.

Findings

Global existence and uniqueness for rational initial data.

Soliton resolution for a dense subset of initial data.

Uniform bounds for higher Sobolev norms.

Abstract

We study the energy-critical half-wave maps equation: \[ \partial_t \mathbf{u} = \mathbf{u} \times |D| \mathbf{u} \] for $\mathbf{u} : [0, T) \times \mathbb{R} \to \mathbb{S}^2$. Our main result establishes the global existence and uniqueness of solutions for all rational initial data $\mathbf{u}_0 : \mathbb{R} \to \mathbb{S}^2$. This demonstrates global well-posedness for a dense subset within the scaling-critical energy space $\dot{H}^{1/2}(\mathbb{R}; \mathbb{S}^2)$. Furthermore, we prove soliton resolution for a dense subset of initial data in the energy space, with uniform bounds for all higher Sobolev norms $\dot{H}^s$ for $s > 0$. Our analysis utilizes the Lax pair structure of the half-wave maps equation on Hardy spaces in combination with an explicit flow formula. Extending these results, we establish global well-posedness for rational initial data (along with a soliton resolution result) for a generalized class of matrix-valued half-wave maps equations with target spaces in the complex Grassmannians $\mathbf{Gr}_k(\mathbb{C}^d)$. Notably, this includes the complex projective spaces $ \mathbb{CP}^{d-1} \cong \mathbf{Gr}_1(\mathbb{C}^d)$ thereby extending the classical case of the target $\mathbb{S}^2 \cong \mathbb{CP}^1$.