Local analytic well-posedness for one-dimensional Vlasov$\unicode{x2013}$Dirac$\unicode{x2013}$Benney-type equations
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866917396728512512 |
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| author | Alves, Nuno J. Markowich, Peter Tzavaras, Athanasios E. |
| author_facet | Alves, Nuno J. Markowich, Peter Tzavaras, Athanasios E. |
| contents | We study a one-dimensional nonlinear Vlasov equation with a local self-consistent force field generated by the density, where the force is given by the spatial derivative of a real-analytic nonlinearity. For small analytic initial data, we prove local-in-time existence and uniqueness of analytic solutions. In particular, this yields a perturbative well-posedness result around the trivial equilibrium. We also give an energy-based representation of weak stationary states and discuss perturbations around spatially homogeneous stationary profiles. The proof relies on a contraction mapping argument in a complete metric space of analytic functions. As a technical byproduct, we establish quantitative composition estimates for analytic nonlinearities in the analytic norms used in the argument. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_04581 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Local analytic well-posedness for one-dimensional Vlasov$\unicode{x2013}$Dirac$\unicode{x2013}$Benney-type equations Alves, Nuno J. Markowich, Peter Tzavaras, Athanasios E. Analysis of PDEs Primary: 35Q83, Secondary: 35F25, 82D10 We study a one-dimensional nonlinear Vlasov equation with a local self-consistent force field generated by the density, where the force is given by the spatial derivative of a real-analytic nonlinearity. For small analytic initial data, we prove local-in-time existence and uniqueness of analytic solutions. In particular, this yields a perturbative well-posedness result around the trivial equilibrium. We also give an energy-based representation of weak stationary states and discuss perturbations around spatially homogeneous stationary profiles. The proof relies on a contraction mapping argument in a complete metric space of analytic functions. As a technical byproduct, we establish quantitative composition estimates for analytic nonlinearities in the analytic norms used in the argument. |
| title | Local analytic well-posedness for one-dimensional Vlasov$\unicode{x2013}$Dirac$\unicode{x2013}$Benney-type equations |
| topic | Analysis of PDEs Primary: 35Q83, Secondary: 35F25, 82D10 |
| url | https://arxiv.org/abs/2412.04581 |