Local Hölder Regularity For Nonlocal Porous Media And Fast Diffusion Equations With General Kernel
Kyeongbae Kim, Ho-Sik Lee, Harsh Prasad
TL;DR
This work proves local Hölder continuity for locally bounded, local weak solutions to a broad class of nonlocal parabolic equations $\partial_t u + (-\Delta)^s(|u|^{m-1}u)=0$ with general kernels. By developing a De Giorgi–Nash–Moser framework tailored to nonlocal, nonlinear diffusion, the authors handle degeneracy and singularity at $u\approx0$ using an intrinsic, solution-dependent geometry and rigorous tail control to manage far-field interactions. The analysis yields an interior $C^{0,\alpha}_{loc}$ regularity result that is robust to initial/boundary data and independent of coefficient regularity, plus a Liouville-type rigidity for bounded global solutions. The results advance the understanding of regularity for fractional porous media and fractional fast diffusion equations with general kernels, providing new interior estimates in the fully nonlinear, local-bounded regime.
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 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.