Sharp Regularity for Weak Solutions to the Porous Medium Equation

Abstract

Let $u$ be a nonnegative, local, weak solution to the porous medium equation for $m\ge 2$ in a space-time cylinder ${\Omega }_T$. Fix a point $(x_o,t_o)\in {\Omega }_T$: if the average \[ a{\buildrel\mbox{def}\over{=}}\frac1{|B_r(x_o)|}\int_{B_r(x_o)}u(x,t_o)\,dx>0, \] then the quantity $|\nabla u^{m-1}|$ is locally bounded in a proper cylinder, whose center lies at time $t_o+a^{1-m}{r}^2$. This implies that in the same cylinder the solution $u$ is Hölder continuous with exponent $\alpha =\frac{1}{m-1}$, which is known to be optimal. Moreover, $u$ presents a sort of instantaneous regularisation, which we quantify.

References in corpus (1)

• Sharp global estimates for local and nonlocal porous medium-type equations in bounded domains