Stefan Sturm | Searcharxiv

arXiv2018

Lower semicontinuous obstacles for the porous medium equation

We deal with the obstacle problem for the porous medium equation in the slow diffusion regime $m>1$. Our main interest is to treat fairly irregular obstacles assuming only boundedness and lower semicontinuity. In particular, the considered obstacles are not regular enough to work with the classical notion of variational solutions, and a different approach is needed. We prove the existence of a solution in the sense of the minimal supersolution lying above the obstacle. As a consequence, we can show that non-negative weak supersolutions to the porous medium equation can be approximated by a sequence of supersolutions which are bounded away from zero.

math.AP↗

arXiv2018

Regularity of weak solutions and supersolutions to the porous medium equation

We study the relations between different regularity assumptions in the definition of weak solutions and supersolutions to the porous medium equation. In particular, we establish the equivalence of the conditions $u^m \in L^2_{\rm loc}(0,T;H^1_{\rm loc}(Ω))$ and $u^\frac{m+1}2 \in L^2_{\rm loc}(0,T;H^1_{\rm loc}(Ω))$ in the definition of weak solutions. Our proof is based on approximation by solutions to obstacle problems.

math.AP↗