Weakly turbulent dynamics on Schwarzschild-AdS black hole spacetimes
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| Format: | Preprint |
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2026
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| _version_ | 1866918445476478976 |
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| author | Kehle, Christoph Moschidis, Georgios |
| author_facet | Kehle, Christoph Moschidis, Georgios |
| contents | In the presence of confinement, small-data solutions to nonlinear dispersive equations can exhibit a gradual energy transfer from low to high frequencies, a mechanism driving the emergence of weakly turbulent dynamics. We show that such a forward energy transfer, manifested as arbitrary inflation of higher order Sobolev norms, occurs for small-data solutions of a quasilinear cubic wave equation on the Schwarzschild-AdS black hole exterior with Dirichlet conditions at infinity, for generic values of the mass parameter. This result is motivated by the question of nonlinear stability or instability of Schwarzschild-AdS as a solution to the Einstein vacuum equations, but the strategy of proof applies to a broader class of backgrounds exhibiting stable trapping of null geodesics. As an application, we obtain the analogous norm inflation statement on $\mathbb R \times \mathbb S^3_+$ for generic perturbations of the round metric on the hemisphere $\mathbb S^3_+$ preserving the trapping structure at the boundary. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_12118 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Weakly turbulent dynamics on Schwarzschild-AdS black hole spacetimes Kehle, Christoph Moschidis, Georgios General Relativity and Quantum Cosmology High Energy Physics - Theory Mathematical Physics Analysis of PDEs In the presence of confinement, small-data solutions to nonlinear dispersive equations can exhibit a gradual energy transfer from low to high frequencies, a mechanism driving the emergence of weakly turbulent dynamics. We show that such a forward energy transfer, manifested as arbitrary inflation of higher order Sobolev norms, occurs for small-data solutions of a quasilinear cubic wave equation on the Schwarzschild-AdS black hole exterior with Dirichlet conditions at infinity, for generic values of the mass parameter. This result is motivated by the question of nonlinear stability or instability of Schwarzschild-AdS as a solution to the Einstein vacuum equations, but the strategy of proof applies to a broader class of backgrounds exhibiting stable trapping of null geodesics. As an application, we obtain the analogous norm inflation statement on $\mathbb R \times \mathbb S^3_+$ for generic perturbations of the round metric on the hemisphere $\mathbb S^3_+$ preserving the trapping structure at the boundary. |
| title | Weakly turbulent dynamics on Schwarzschild-AdS black hole spacetimes |
| topic | General Relativity and Quantum Cosmology High Energy Physics - Theory Mathematical Physics Analysis of PDEs |
| url | https://arxiv.org/abs/2604.12118 |