Regularity of Hölder continuous solutions of a modified critical porous media equation

Abstract

In this paper, we study the regularity of weak solutions of a modified critical dissipative porous media equation. by using Besov space techniques combined with energy estimates, we prove that Holder continuous weak solutions to this system are smooth solutions. The porous media equation is a dissipative transport-diffusion equation with non-local divergence-free velocity field. It shares many similarities with the surface quasi-geostrophic equation. From a mathematical view, the porous media equation is somewhat a generalization of the surface quasi-geostrophic equation. In this paper, we study the regularity of weak solutions of a 3-D modified critical dissipative porous media equation. By using Besov space techniques combined with classical energy estimates, we prove that Holder continuous weak solutions to this system are smooth solutions. In particular, by applying the same method, we obtain that this result holds true for the 3-D modified critical surface quasi-geostrophic equation.