Regularity and Uniqueness for a Model of Active Particles with Angle-Averaged Diffusions
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866908521555034112 |
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| author | Alasio, Luca Schulz, Simon |
| author_facet | Alasio, Luca Schulz, Simon |
| contents | We study the regularity and uniqueness of weak solutions of a degenerate parabolic equation, arising as the limit of a stochastic lattice model of self-propelled particles. The angle-average of the solution appears as a coefficient in the diffusive and drift terms, making the equation nonlocal. We prove that, under unrestrictive non-degeneracy assumptions on the initial data, weak solutions are smooth for positive times. Our method rests on deriving a drift-diffusion equation for a particular function of the angle-averaged density and applying De Giorgi's method to show that the original equation is uniformly parabolic for positive times. We employ a Galerkin approximation to justify rigorously the passage from divergence to non-divergence form of the equation, which yields improved estimates by exploiting a cancellation. By imposing stronger constraints on the initial data, we prove the uniqueness of the weak solution, which relies on Duhamel's principle and gradient estimates for the periodic heat kernel to derive $L^\infty$ estimates for the angle-averaged density. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_11488 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Regularity and Uniqueness for a Model of Active Particles with Angle-Averaged Diffusions Alasio, Luca Schulz, Simon Analysis of PDEs 35B65, 35K55, 35Q70, 35Q84 We study the regularity and uniqueness of weak solutions of a degenerate parabolic equation, arising as the limit of a stochastic lattice model of self-propelled particles. The angle-average of the solution appears as a coefficient in the diffusive and drift terms, making the equation nonlocal. We prove that, under unrestrictive non-degeneracy assumptions on the initial data, weak solutions are smooth for positive times. Our method rests on deriving a drift-diffusion equation for a particular function of the angle-averaged density and applying De Giorgi's method to show that the original equation is uniformly parabolic for positive times. We employ a Galerkin approximation to justify rigorously the passage from divergence to non-divergence form of the equation, which yields improved estimates by exploiting a cancellation. By imposing stronger constraints on the initial data, we prove the uniqueness of the weak solution, which relies on Duhamel's principle and gradient estimates for the periodic heat kernel to derive $L^\infty$ estimates for the angle-averaged density. |
| title | Regularity and Uniqueness for a Model of Active Particles with Angle-Averaged Diffusions |
| topic | Analysis of PDEs 35B65, 35K55, 35Q70, 35Q84 |
| url | https://arxiv.org/abs/2501.11488 |