Regularity of weak solutions and supersolutions to the porous medium equation

Abstract

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 ∈ L 2 loc (0, T ; H 1 loc (Ω)) and u m+1 2 ∈ L 2 loc (0, T ; H 1 loc (Ω)) in the definition of weak solutions. Our proof is based on approximation by solutions to obstacle problems.

Supercaloric functions for the porous medium equation

“…In order to obtain appropriate Caccioppoli type energy estimates it is convenient to impose the Sobolev space assumption on u m+1 2 instead of u m in the definition above. According to the recent result in [2], this does not make any difference for locally bounded functions.…”

Hölder regularity for porous medium systems

“…5 The notion of solution may generate discrepancies, as it is not entirely clear even in the scalar case. An interesting study on the equivalence of different notions of solution for the prototype porous medium equation is carried out in [1].…”

“…. This becomes a non-degenerate, diagonal system about w. According to [12], there exist α 1 ∈ (0, 1) and γ > 1 depending only on the data, such that for all 0 < r < 1 2 , max 1≤i≤n ess osc…”

Boundary Regularity for the Porous Medium Equation

“…Such a choice seems more natural in a number of applications, but it seemingly introduces the extra difficulty that two different notions of solutions are needed, according to whether m ≤ 1 or m ≥ 1. However, it has recently been proved by Bögelein-Lehtelä-Sturm [15,Theorem 1.2], that for m ≥ 1 the two notions are equivalent.…”