Local Hölder Regularity For Nonlocal Porous Media And Fast Diffusion Equations With General Kernel
Year: 2025
Type: preprint
Abstract: 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 00 \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.
Source: arXiv (Cornell University)
Authors: Kim, Kyeongbae, Lee, Ho-Sik, Prasad, Harsh
Institutions: Seoul National University, Bielefeld University
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W6979249054
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