Global well-posedness and relaxation for solutions of the Fokker-Planck-Alignment equations
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914335486377984 |
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| author | Shvydkoy, R. |
| author_facet | Shvydkoy, R. |
| contents | In this paper we prove global existence of weak solutions, their regularization, and relaxation for large data for a broad class of Fokker-Planck-Alignment models which appear in collective dynamics. The main feature of these results, as opposed to previously known ones, is the lack of regularity or no-vacuum requirements on the initial data. With a particular application to the classical kinetic Cucker-Smale model, we demonstrate that any bounded data with finite higher moment, $f_0 \in L^1(1+ |v|^q) \cap L^\infty$, $q \geq n+4$, gives rise to a global instantly smooth solution, satisfying entropy equality and relaxing exponentially fast.
The results are achieved through the use of a new thickness-based renormalization procedure, which circumvents the problem of degenerate diffusion in non-perturbative regime. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_20294 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Global well-posedness and relaxation for solutions of the Fokker-Planck-Alignment equations Shvydkoy, R. Analysis of PDEs In this paper we prove global existence of weak solutions, their regularization, and relaxation for large data for a broad class of Fokker-Planck-Alignment models which appear in collective dynamics. The main feature of these results, as opposed to previously known ones, is the lack of regularity or no-vacuum requirements on the initial data. With a particular application to the classical kinetic Cucker-Smale model, we demonstrate that any bounded data with finite higher moment, $f_0 \in L^1(1+ |v|^q) \cap L^\infty$, $q \geq n+4$, gives rise to a global instantly smooth solution, satisfying entropy equality and relaxing exponentially fast. The results are achieved through the use of a new thickness-based renormalization procedure, which circumvents the problem of degenerate diffusion in non-perturbative regime. |
| title | Global well-posedness and relaxation for solutions of the Fokker-Planck-Alignment equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2412.20294 |