Nonlocal approximation of nonlinear diffusion equations
We show that degenerate nonlinear diffusion equations can be asymptotically obtained as a limit from a class of nonlocal partial differential equations. The nonlocal equations are obtained as gradient flows of interaction-like energies approximating the internal energy. We construct weak solutions a...
| المؤلفون الرئيسيون: | , , |
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| التنسيق: | Journal article |
| اللغة: | English |
| منشور في: |
Springer Nature
2024
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| _version_ | 1826314515510198272 |
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| author | Carrillo, JA Esposito, A Wu, JS-H |
| author_facet | Carrillo, JA Esposito, A Wu, JS-H |
| author_sort | Carrillo, JA |
| collection | OXFORD |
| description | We show that degenerate nonlinear diffusion equations can be asymptotically obtained as a limit from a class of nonlocal partial differential equations. The nonlocal equations are obtained as gradient flows of interaction-like energies approximating the internal energy. We construct weak solutions as the limit of a (sub)sequence of weak measure solutions by using the Jordan-Kinderlehrer-Otto scheme from the context of 2-Wasserstein gradient flows. Our strategy allows to cover the porous medium equation, for the general slow diffusion case, extending previous results in the literature. As a byproduct of our analysis, we provide a qualitative particle approximation. |
| first_indexed | 2024-09-25T04:35:16Z |
| format | Journal article |
| id | oxford-uuid:90aee5f5-a09d-48a7-ad4e-812b15de1f6d |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-09-25T04:35:16Z |
| publishDate | 2024 |
| publisher | Springer Nature |
| record_format | dspace |
| spelling | oxford-uuid:90aee5f5-a09d-48a7-ad4e-812b15de1f6d2024-09-17T11:40:22ZNonlocal approximation of nonlinear diffusion equationsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:90aee5f5-a09d-48a7-ad4e-812b15de1f6dEnglishSymplectic ElementsSpringer Nature2024Carrillo, JAEsposito, AWu, JS-HWe show that degenerate nonlinear diffusion equations can be asymptotically obtained as a limit from a class of nonlocal partial differential equations. The nonlocal equations are obtained as gradient flows of interaction-like energies approximating the internal energy. We construct weak solutions as the limit of a (sub)sequence of weak measure solutions by using the Jordan-Kinderlehrer-Otto scheme from the context of 2-Wasserstein gradient flows. Our strategy allows to cover the porous medium equation, for the general slow diffusion case, extending previous results in the literature. As a byproduct of our analysis, we provide a qualitative particle approximation. |
| spellingShingle | Carrillo, JA Esposito, A Wu, JS-H Nonlocal approximation of nonlinear diffusion equations |
| title | Nonlocal approximation of nonlinear diffusion equations |
| title_full | Nonlocal approximation of nonlinear diffusion equations |
| title_fullStr | Nonlocal approximation of nonlinear diffusion equations |
| title_full_unstemmed | Nonlocal approximation of nonlinear diffusion equations |
| title_short | Nonlocal approximation of nonlinear diffusion equations |
| title_sort | nonlocal approximation of nonlinear diffusion equations |
| work_keys_str_mv | AT carrilloja nonlocalapproximationofnonlineardiffusionequations AT espositoa nonlocalapproximationofnonlineardiffusionequations AT wujsh nonlocalapproximationofnonlineardiffusionequations |