[Paper Review] Sharp Regularity for Weak Solutions to the Porous Medium Equation

Ugo Gianazza, Juhana Siljander|arXiv (Cornell University)|Jul 23, 2016

Nonlinear Partial Differential Equations17 references3 citations

TL;DR

This paper establishes sharp Hölder regularity for nonnegative weak solutions to the porous medium equation with $ m \geq 2 $, proving that $ |\nabla u^{m-1}| $ is locally bounded in a time-shifted cylinder centered at $ t_0 + a^{1-m}r^2 $, where $ a $ is the spatial average of $ u $ at time $ t_0 $. This yields optimal Hölder continuity of $ u $ with exponent $ \alpha = \frac{1}{m-1} $, quantifying instantaneous regularization even after potential singularities at the free boundary.

ABSTRACT

Let $u$ be a nonnegative, local, weak solution to the porous medium equation for $m\ge2$ in a space-time cylinder $Ω_T$. Fix a point $(x_o,t_o)\inΩ_T$: 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 $| abla u^{m-1}|$ 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 $α=\frac1{m-1}$, which is known to be optimal. Moreover, $u$ presents a sort of instantaneous regularisation, which we quantify.

Motivation & Objective

• To establish sharp regularity estimates for weak solutions to the porous medium equation with $ m \geq 2 $, particularly focusing on the optimal Hölder exponent.
• To quantify the instantaneous regularization of solutions after potential singularities at the free boundary, even in the presence of Aronson–Graveleau-type counter-examples.
• To prove local boundedness of $ |\nabla u^{m-1}| $ in a time-shifted parabolic cylinder, which implies optimal Hölder continuity of $ u $.
• To provide purely interior estimates without requiring global or initial data assumptions, contrasting with prior waiting-time results in the literature.
• To lay a foundation for future regularity analysis of the free boundary by establishing sharp gradient estimates from interior arguments.

Proposed method

• Use of intrinsic Harnack and weak Harnack inequalities to control solution behavior in parabolic cylinders.
• Application of a quantitative $ L^\infty $ estimate to bound the solution in terms of its spatial average at a fixed time.
• Construction of a comparison function $ u_{\epsilon,r} $ via regularization and truncation to control the solution from above.
• Employment of a time-shifted cylinder $ (x_0,t_0) + Q_{r/2}^-(\bar{\theta}) $ with $ \bar{\theta} = a^{1-m} $, where $ a $ is the average of $ u $ at $ t_0 $.
• Use of the definition $ \nabla u = \frac{2}{m+1} \mathbf{1}_{\{u>0\}} u^{\frac{1-m}{2}} \nabla u^{\frac{m+1}{2}} $ to handle degeneracy in the equation.
• Limiting argument as $ \epsilon \to 0 $ to pass from regularized solutions to the original weak solution, preserving the estimates.

Experimental results

Research questions

• RQ1 Does the solution $ u $ to the porous medium equation with $ m \geq 2 $ achieve the optimal Hölder continuity exponent $ \alpha = \frac{1}{m-1} $ in the interior, even after singularities at the free boundary?
• RQ2 Can the gradient $ |\nabla u^{m-1}| $ be locally bounded in a time-shifted parabolic cylinder centered at $ t_0 + a^{1-m}r^2 $, where $ a $ is the spatial average of $ u $ at $ t_0 $?
• RQ3 Is there a quantifiable form of instantaneous regularization for weak solutions, even in the presence of Aronson–Graveleau-type singular behavior?
• RQ4 Can sharp regularity estimates be derived from purely interior arguments, independent of initial or global data?
• RQ5 Does the sharp regularity result support the regularity of the free boundary in the porous medium equation?

Key findings

• The gradient $ |\nabla u^{m-1}| $ is locally bounded in the cylinder $ (x_0,t_0) + Q_{r/2}^-(M^{1-m}) $, where $ M $ is the spatial average of $ u $ at $ t_0 $, with $ M \geq \gamma r^{1/(m-1)} $.
• The solution $ u $ is Hölder continuous with exponent $ \alpha = \frac{1}{m-1} $ in the same cylinder, which is optimal and sharp.
• The estimate $ \sup_{(x_0,t_0)+Q_{r/2}^-(M^{1-m})} u(x,t) \leq \kappa(\gamma + 2)r^{1/(m-1)} $ holds for a constant $ \kappa $ depending only on $ \{m,N\} $.
• The result quantifies instantaneous regularization: even if the solution is singular at the free boundary, regularity is restored immediately in the time-shifted cylinder.
• The proof relies on intrinsic Harnack inequalities, quantitative $ L^\infty $ estimates, and comparison principles, with no dependence on initial data or global behavior.
• The method extends to doubly nonlinear equations of the form $ u_t - \operatorname{div}(m|u|^{m-1}|\nabla u|^{p-2}\nabla u) = 0 $, suggesting the optimal Hölder exponent may be $ \alpha = \frac{p-1}{m+p-3} $.