Work

Local Hölder regularity for nonlocal porous media and fast diffusion equations with general kernel

Year: 2026

Type: article

Abstract: We show that locally bounded, local weak solutions to certain nonlocal, nonlinear diffusion equations modelled on the fractional porous media and fast diffusion equations given by $$\begin{aligned} \partial _t u + (-\Delta )^s(|u|^{m-1}u) = 0 \quad \text{ for } \quad 00 \end{aligned}$$ ∂ t u + ( - Δ ) s ( | u | m - 1 u ) = 0 for 0 < s < 1 and m > 0 are locally Hölder continuous. We work with bounded, measurable kernels and provide the corresponding $$L^{\infty }_{loc} \rightarrow C^{0,\alpha }_{loc}$$ L loc ∞ → C loc 0 , α 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: Calculus of Variations and Partial Differential Equations

DOI: 10.1007/s00526-026-03306-4

Authors: Kyeongbae Kim, Ho-Sik Lee, Harsh Prasad

Institutions: University of Bonn, Bielefeld University

Language: English

Topic: Nonlinear Partial Differential Equations

Subfield: Applied Mathematics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,190

Funder: Deutsche Forschungsgemeinschaft