Global higher integrability of weak solutions of porous medium systems

Kristian Moring; Christoph Scheven; Sebastian Schwarzacher; Thomas Singer

doi:10.3934/cpaa.2020069

Article Contents

2020, Volume 19, Issue 3: 1697-1745. Doi: 10.3934/cpaa.2020069

This issue Previous Article Homogenization of a locally periodic time-dependent domain Next Article Bending-torsion moments in thin multi-structures in the context of nonlinear elasticity

Global higher integrability of weak solutions of porous medium systems

1.
Department of Mathematics and Systems Analysis, Aalto University, P. O. Box 11100, FI-00076 Aalto, Finland
2.
Fakultät für Mathematik, Universität Duisburg-Essen, Thea-Leymann-Str. 9, 45127 Essen, Germany
3.
Katedra matematické analyźy, Matematicko-fyzikální fakulta Univerzity Karlovy, Sokolovská 83,186 75 Praha 8, Czech Republic
4.
Department Mathematik, Friedrich-Alexander-Universität Erlangen-Nürnberg, Cauerstrasse 11, 91058 Erlangen, Germany

^* Corresponding author
^* Corresponding author

Received: March 2019

Revised: August 2019

Published online: March 2020

K. Moring has been supported by the Magnus Ehrnrooth foundation. T. Singer has been supported by the DFG-Project SI 2464/1-1 ``Highly nonlinear evolutionary problems".

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Abstract

We establish higher integrability up to the boundary for the gradient of solutions to porous medium type systems, whose model case is given by

$ \begin{equation*} \partial_t u-\Delta(|u|^{m-1}u) = \mathrm{div}\,F\,, \end{equation*} $

where $ m>1 $. More precisely, we prove that under suitable assumptions the spatial gradient $ D(|u|^{m-1}u) $ of any weak solution is integrable to a larger power than the natural power $ 2 $. Our analysis includes both the case of the lateral boundary and the initial boundary.

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Mathematics Subject Classification: 35B65, 35K65, 35K40, 35K55.

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