Nonlinear chaotic Vlasov equations
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866929410500722688 |
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| author | Chaubet, Yann Han-Kwan, Daniel Rivière, Gabriel |
| author_facet | Chaubet, Yann Han-Kwan, Daniel Rivière, Gabriel |
| contents | In this article, we study nonlinear Vlasov equations with a smooth interaction kernel on a compact manifold without boundary where the geodesic flow exhibits strong chaotic behavior, known as the Anosov property. We show that, for small initial data with finite regularity and supported away from the null section, there exist global solutions to the nonlinear Vlasov equations which weakly converge to an equilibrium of the free transport equation, and whose potential strongly converges to zero, both with exponential speed. Central to our approach are microlocal anisotropic Sobolev spaces, originally developed for studying Pollicott-Ruelle resonances, that we further refine to deal with the geometry of the full cotangent bundle, which paves the way to the analysis of nonlinear Vlasov equations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_04426 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Nonlinear chaotic Vlasov equations Chaubet, Yann Han-Kwan, Daniel Rivière, Gabriel Analysis of PDEs Dynamical Systems 35Q83, 37D40, 58J45 In this article, we study nonlinear Vlasov equations with a smooth interaction kernel on a compact manifold without boundary where the geodesic flow exhibits strong chaotic behavior, known as the Anosov property. We show that, for small initial data with finite regularity and supported away from the null section, there exist global solutions to the nonlinear Vlasov equations which weakly converge to an equilibrium of the free transport equation, and whose potential strongly converges to zero, both with exponential speed. Central to our approach are microlocal anisotropic Sobolev spaces, originally developed for studying Pollicott-Ruelle resonances, that we further refine to deal with the geometry of the full cotangent bundle, which paves the way to the analysis of nonlinear Vlasov equations. |
| title | Nonlinear chaotic Vlasov equations |
| topic | Analysis of PDEs Dynamical Systems 35Q83, 37D40, 58J45 |
| url | https://arxiv.org/abs/2407.04426 |