Drift-diffusion equations with saturation
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866917087092408320 |
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| author | Carrillo, José Antonio Fernández-Jiménez, Alejandro Gómez-Castro, David |
| author_facet | Carrillo, José Antonio Fernández-Jiménez, Alejandro Gómez-Castro, David |
| contents | We focus on a family of nonlinear continuity equations for the evolution of a non-negative density $ρ$ with a continuous and compactly supported nonlinear mobility $\mathrm{m}(ρ)$ not necessarily concave. The velocity field is the negative gradient of the variation of a free energy including internal and confinement energy terms. Problems with compactly supported mobility are often called saturation problems since the values of the density are constrained below a maximal value. Taking advantage of a family of approximating problems, we show the existence of $C_0$-semigroups of $L^1$ contractions. We study the $ω$-limit of the problem, its most relevant properties, and the appearance of free boundaries in the long-time behaviour. This problem has a formal gradient-flow structure, and we discuss the local/global minimisers of the corresponding free energy in the natural topology related to the set of initial data for the $L^\infty$-constrained gradient flow of probability densities. Furthermore, we analyse a structure preserving implicit finite-volume scheme and discuss its convergence and long-time behaviour. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_10040 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Drift-diffusion equations with saturation Carrillo, José Antonio Fernández-Jiménez, Alejandro Gómez-Castro, David Analysis of PDEs Numerical Analysis 35K55, 35K65, 35B40, 65M08, 35Q70, 35Q92, 47H20 We focus on a family of nonlinear continuity equations for the evolution of a non-negative density $ρ$ with a continuous and compactly supported nonlinear mobility $\mathrm{m}(ρ)$ not necessarily concave. The velocity field is the negative gradient of the variation of a free energy including internal and confinement energy terms. Problems with compactly supported mobility are often called saturation problems since the values of the density are constrained below a maximal value. Taking advantage of a family of approximating problems, we show the existence of $C_0$-semigroups of $L^1$ contractions. We study the $ω$-limit of the problem, its most relevant properties, and the appearance of free boundaries in the long-time behaviour. This problem has a formal gradient-flow structure, and we discuss the local/global minimisers of the corresponding free energy in the natural topology related to the set of initial data for the $L^\infty$-constrained gradient flow of probability densities. Furthermore, we analyse a structure preserving implicit finite-volume scheme and discuss its convergence and long-time behaviour. |
| title | Drift-diffusion equations with saturation |
| topic | Analysis of PDEs Numerical Analysis 35K55, 35K65, 35B40, 65M08, 35Q70, 35Q92, 47H20 |
| url | https://arxiv.org/abs/2410.10040 |