Well-posedness of the Boltzmann and Landau Equations in Critical Spaces
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912592451076096 |
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| author | Chen, Ke Nguyen, Quoc-Hung Yang, Tong |
| author_facet | Chen, Ke Nguyen, Quoc-Hung Yang, Tong |
| contents | This paper investigates the well-posedness of the inhomogeneous Boltzmann and Landau equations in critical function spaces, a fundamental open problem in kinetic theory. We develop a new analytical framework to establish local well-posedness near a global Maxwellian for both equations, under the assumption that the initial perturbation is small in a critical norm. A major contribution lies in the introduction of a novel anisotropic norm adapted to the intrinsic scaling invariance of the equations, which provides precise control over the high-frequency behavior of solutions. By leveraging the regularizing effect and a decomposition of the linearized collision operator, we further extend the local solution globally in time and establish pointwise decay estimates. Our work not only resolves a fundamental issue in the theory of kinetic equations in critical spaces, but also provides a new approach applicable to a broader class of kinetic models. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_14845 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Well-posedness of the Boltzmann and Landau Equations in Critical Spaces Chen, Ke Nguyen, Quoc-Hung Yang, Tong Analysis of PDEs This paper investigates the well-posedness of the inhomogeneous Boltzmann and Landau equations in critical function spaces, a fundamental open problem in kinetic theory. We develop a new analytical framework to establish local well-posedness near a global Maxwellian for both equations, under the assumption that the initial perturbation is small in a critical norm. A major contribution lies in the introduction of a novel anisotropic norm adapted to the intrinsic scaling invariance of the equations, which provides precise control over the high-frequency behavior of solutions. By leveraging the regularizing effect and a decomposition of the linearized collision operator, we further extend the local solution globally in time and establish pointwise decay estimates. Our work not only resolves a fundamental issue in the theory of kinetic equations in critical spaces, but also provides a new approach applicable to a broader class of kinetic models. |
| title | Well-posedness of the Boltzmann and Landau Equations in Critical Spaces |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2509.14845 |