Mode stability of blow-up for wave maps in the absence of symmetry
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| Format: | Preprint |
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2025
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| _version_ | 1866916642658713600 |
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| author | Weissenbacher, Max Koch, Herbert Donninger, Roland |
| author_facet | Weissenbacher, Max Koch, Herbert Donninger, Roland |
| contents | The wave maps equation in three spatial dimensions with a spherical target admits an explicit blow-up solution. Numerical studies suggest this solution captures the generic blow-up behaviour in the backward light cone of the singularity. In this work, we establish the mode stability of this blow-up solution in the backward light cone of the blow-up point without any assumptions on the symmetries of the perturbation. We classify all smooth mode solutions for growth rates $λ$ with $\mathrm{Re} \, λ\geq 0$ and demonstrate that the blow-up solution is stable up to the mode solutions arising from the symmetry group of the wave maps equation. Our proof relies on a decomposition of the linearised wave maps equation into a tractable system of symmetry-equivariant ordinary differential equations (ODEs), utilising the representation theory of the stabiliser of the blow-up solution. We then use the quasi-solution method of Costin-Donninger-Glogić to show the absence of non-zero smooth solutions for the resulting system of ODEs. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_02632 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Mode stability of blow-up for wave maps in the absence of symmetry Weissenbacher, Max Koch, Herbert Donninger, Roland Analysis of PDEs Mathematical Physics Classical Analysis and ODEs 35L71 (Primary), 35B44 (Secondary), 35B35 (Secondary), 35Q75 (Secondary) The wave maps equation in three spatial dimensions with a spherical target admits an explicit blow-up solution. Numerical studies suggest this solution captures the generic blow-up behaviour in the backward light cone of the singularity. In this work, we establish the mode stability of this blow-up solution in the backward light cone of the blow-up point without any assumptions on the symmetries of the perturbation. We classify all smooth mode solutions for growth rates $λ$ with $\mathrm{Re} \, λ\geq 0$ and demonstrate that the blow-up solution is stable up to the mode solutions arising from the symmetry group of the wave maps equation. Our proof relies on a decomposition of the linearised wave maps equation into a tractable system of symmetry-equivariant ordinary differential equations (ODEs), utilising the representation theory of the stabiliser of the blow-up solution. We then use the quasi-solution method of Costin-Donninger-Glogić to show the absence of non-zero smooth solutions for the resulting system of ODEs. |
| title | Mode stability of blow-up for wave maps in the absence of symmetry |
| topic | Analysis of PDEs Mathematical Physics Classical Analysis and ODEs 35L71 (Primary), 35B44 (Secondary), 35B35 (Secondary), 35Q75 (Secondary) |
| url | https://arxiv.org/abs/2503.02632 |