Let $u$ be a nonnegative, local, weak solution to the porous medium equation \[ \partial_t u-\Delta u^m=0 \] for $m\ge2$ in a space-time cylinder $\Om_T=\Om\times(0,T]$. Fix a point $\pto\in\Om_T$: if the average \[ a\df=\dashint_{B_r(x_o)}u(x,t_o)\,dx>0, \] 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 $\al=\frac1{m-1}$, which is known to be optimal}. Moreover, $u$ presents a sort of instantaneous regularization, which we discuss.
Local bounds of the gradient of weak solutions to the porous medium equation
Gianazza, Ugo
;Siljander, Juhana
2023-01-01
Abstract
Let $u$ be a nonnegative, local, weak solution to the porous medium equation \[ \partial_t u-\Delta u^m=0 \] for $m\ge2$ in a space-time cylinder $\Om_T=\Om\times(0,T]$. Fix a point $\pto\in\Om_T$: if the average \[ a\df=\dashint_{B_r(x_o)}u(x,t_o)\,dx>0, \] 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 $\al=\frac1{m-1}$, which is known to be optimal}. Moreover, $u$ presents a sort of instantaneous regularization, which we discuss.File in questo prodotto:
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