Local Hölder Regularity For Nonlocal Porous Media And Fast Diffusion Equations With General Kernel
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866912340827439104 |
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| author | Kim, Kyeongbae Lee, Ho-Sik Prasad, Harsh |
| author_facet | Kim, Kyeongbae Lee, Ho-Sik Prasad, Harsh |
| contents | We show that locally bounded, local weak solutions to certain nonlocal, nonlinear diffusion equations modeled on the fractional porous media and fast diffusion equations given by \begin{align*} \partial_t u + (-Δ)^s(|u|^{m-1}u) = 0 \quad \mbox{ for } \quad 0<s<1 \quad\text{and}\quad m>0 \end{align*} are locally Hölder continuous. We work with bounded, measurable kernels and provide the corresponding $L^{\infty}_{loc} \rightarrow C^{0,α}_{loc}$ De Giorgi-Nash-Moser theory for the equation via a delicate analysis of the set of singularity/degeneracy in a geometry dictated by the solution itself and a careful analysis of far-off effects. In particular, our results are in the spirit of interior regularity, requiring the equation to hold only locally, and thus are new even for positive solutions of the equation with constant coefficients. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_15735 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Local Hölder Regularity For Nonlocal Porous Media And Fast Diffusion Equations With General Kernel Kim, Kyeongbae Lee, Ho-Sik Prasad, Harsh Analysis of PDEs 35R09, 35B65, 47G20, 35K55 We show that locally bounded, local weak solutions to certain nonlocal, nonlinear diffusion equations modeled on the fractional porous media and fast diffusion equations given by \begin{align*} \partial_t u + (-Δ)^s(|u|^{m-1}u) = 0 \quad \mbox{ for } \quad 0<s<1 \quad\text{and}\quad m>0 \end{align*} are locally Hölder continuous. We work with bounded, measurable kernels and provide the corresponding $L^{\infty}_{loc} \rightarrow C^{0,α}_{loc}$ De Giorgi-Nash-Moser theory for the equation via a delicate analysis of the set of singularity/degeneracy in a geometry dictated by the solution itself and a careful analysis of far-off effects. In particular, our results are in the spirit of interior regularity, requiring the equation to hold only locally, and thus are new even for positive solutions of the equation with constant coefficients. |
| title | Local Hölder Regularity For Nonlocal Porous Media And Fast Diffusion Equations With General Kernel |
| topic | Analysis of PDEs 35R09, 35B65, 47G20, 35K55 |
| url | https://arxiv.org/abs/2504.15735 |