Global well-posedness and soliton resolution for the half-wave maps equation with rational data

Gérard, Patrick  ; Lenzmann, Enno 

Forum of Mathematics, Sigma, Tome 13 (2025) no. 1, p. e190

Résumé

We study the energy-critical half-wave maps equation: $$\begin{align*}\partial_t \mathbf{u} = \mathbf{u} \times |D| \mathbf{u} \end{align*}$$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 $ \mathsf {Gr}_k(\mathbb {C}^d) $. Notably, this includes the complex projective spaces $ \mathbb {CP}^{d-1} \cong \mathsf {Gr}_1(\mathbb {C}^d) $ thereby extending the classical case of the target $\mathbb {S}^2 \cong \mathbb {CP}^1 $.

Détail

DOI : 10.1017/fms.2025.10136

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