Regularity of weak solutions and supersolutions to the porous medium equation

Regularity of weak solutions and supersolutions to the porous medium equation

We investigate 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 um∈Lloc2(0,T;Hloc1(Ω)) and um+12∈Lloc2(0,T;Hloc1(Ω)) in the definition of weak sol...

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Veröffentlicht in:Nonlinear analysis 2019-08, Vol.185, p.49-67
Hauptverfasser: Bögelein, Verena, Lehtelä, Pekka, Sturm, Stefan
Format: Artikel
Sprache:eng
Schlagworte:
Gradient estimates
Obstacle problem
Porous media
Porous medium equation
Regularity
Weak solutions
Weak supersolutions
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Zusammenfassung:We investigate 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 um∈Lloc2(0,T;Hloc1(Ω)) and um+12∈Lloc2(0,T;Hloc1(Ω)) in the definition of weak solutions. Our proof is based on approximation by solutions to obstacle problems.
ISSN:0362-546X
1873-5215
DOI:10.1016/j.na.2018.10.019