Convergence of a finite-volume scheme for aggregation-diffusion equations with saturation
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911057813962752 |
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| author | Gómez-Castro, David |
| author_facet | Gómez-Castro, David |
| contents | In [Bailo, Carrillo, Hu. SIAM J. Appl. Math. 2023] the authors introduce a finite-volume method for aggregation-diffusion equations with non-linear mobility. In this paper we prove convergence of this method using an Aubin--Simons compactness theorem due to Gallouët and Latché. We use suitable discrete $H^1$ and $W^{-1,1}$ discrete norms. We provide two convergence results. A first result shows convergence with general entropies ($U$) (including singular and degenerate) if the initial datum does not have free boundaries, the mobility is Lipschitz, and the confinement ($V$) and aggregation ($K$) potentials are $W^{2,\infty}_0$. A second result shows convergence when the initial datum has free boundaries, mobility is just continuous, and $V$ and $K$ are $W^{1,\infty}$, but under the assumption that the entropy $U$ is $C^1$ and strictly convex. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_11132 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Convergence of a finite-volume scheme for aggregation-diffusion equations with saturation Gómez-Castro, David Numerical Analysis Analysis of PDEs 65M08, 35Q70, 35Q92, 45K05 In [Bailo, Carrillo, Hu. SIAM J. Appl. Math. 2023] the authors introduce a finite-volume method for aggregation-diffusion equations with non-linear mobility. In this paper we prove convergence of this method using an Aubin--Simons compactness theorem due to Gallouët and Latché. We use suitable discrete $H^1$ and $W^{-1,1}$ discrete norms. We provide two convergence results. A first result shows convergence with general entropies ($U$) (including singular and degenerate) if the initial datum does not have free boundaries, the mobility is Lipschitz, and the confinement ($V$) and aggregation ($K$) potentials are $W^{2,\infty}_0$. A second result shows convergence when the initial datum has free boundaries, mobility is just continuous, and $V$ and $K$ are $W^{1,\infty}$, but under the assumption that the entropy $U$ is $C^1$ and strictly convex. |
| title | Convergence of a finite-volume scheme for aggregation-diffusion equations with saturation |
| topic | Numerical Analysis Analysis of PDEs 65M08, 35Q70, 35Q92, 45K05 |
| url | https://arxiv.org/abs/2507.11132 |