Sobolev instability for perturbations of periodic transport equations
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918164066992128 |
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| author | Rivière, Gabriel Rotolo, Maria Teresa |
| author_facet | Rivière, Gabriel Rotolo, Maria Teresa |
| contents | We consider linear and time-dependent perturbations of periodic transport equations on the two-dimensional torus. For generic perturbations, we prove the existence of a large class of initial data whose Sobolev norms diverge exponentially fast. In higher dimensions, this remains true under a Morse-Smale assumption on the resonant part of the perturbation. In both cases, this is achieved by a normal form procedure and by studying Sobolev instabilities for time-dependent perturbations of Morse-Smale transport equations. The latter are analyzed on general compact manifolds using techniques from microlocal analysis and hyperbolic dynamics. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_17350 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sobolev instability for perturbations of periodic transport equations Rivière, Gabriel Rotolo, Maria Teresa Analysis of PDEs Dynamical Systems We consider linear and time-dependent perturbations of periodic transport equations on the two-dimensional torus. For generic perturbations, we prove the existence of a large class of initial data whose Sobolev norms diverge exponentially fast. In higher dimensions, this remains true under a Morse-Smale assumption on the resonant part of the perturbation. In both cases, this is achieved by a normal form procedure and by studying Sobolev instabilities for time-dependent perturbations of Morse-Smale transport equations. The latter are analyzed on general compact manifolds using techniques from microlocal analysis and hyperbolic dynamics. |
| title | Sobolev instability for perturbations of periodic transport equations |
| topic | Analysis of PDEs Dynamical Systems |
| url | https://arxiv.org/abs/2510.17350 |