Sharp Regularity for Weak Solutions to the Porous Medium Equation

Let \(u\) be a nonnegative, local, weak solution to the porous medium equation for \(m\ge2\) 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 H\"older continuous with exponent \(\alpha=\frac1{m-1}\), which is known to be optimal. Moreover, \(u\) presents a sort of instantaneous regularisation, which we quantify.