Global Well-Posedness and Soliton Resolution for the Half-Wave Maps Equation with Rational Data
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| author | Gérard, Patrick Lenzmann, Enno |
| author_facet | Gérard, Patrick Lenzmann, Enno |
| contents | 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 $\mathbb{S}^2 \cong \mathbb{CP}^1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_03351 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Global Well-Posedness and Soliton Resolution for the Half-Wave Maps Equation with Rational Data Gérard, Patrick Lenzmann, Enno Analysis of PDEs Mathematical Physics Exactly Solvable and Integrable Systems 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 $\mathbb{S}^2 \cong \mathbb{CP}^1$. |
| title | Global Well-Posedness and Soliton Resolution for the Half-Wave Maps Equation with Rational Data |
| topic | Analysis of PDEs Mathematical Physics Exactly Solvable and Integrable Systems |
| url | https://arxiv.org/abs/2412.03351 |