The higher integrability of weak solutions of porous medium systems

Bögelein, Verena; Duzaar, Frank; Korte, Riikka; Scheven, Christoph

doi:10.1515/anona-2017-0270

Cited by 41 publications

(53 citation statements)

References 25 publications

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“…The proof, however, uses the method of expansion of positivity and therefore can not be extended to signed solutions, porous medium type systems and the fast diffusion range. A simpler and more flexible proof, which does not rely on the expansion of positivity and which covers both signed solutions and porous medium systems is given in [3]. Finally, in [2] the higher integrability is shown for doubly nonlinear parabolic systems, whose prototype is…”

Section: Introduction and Resultsmentioning

confidence: 99%

Higher integrability for the singular porous medium system

Bögelein

Duzaar

Scheven

2019

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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In this paper we establish in the fast diffusion range the higher integrability of the spatial gradient of weak solutions to porous medium systems. The result comes along with an explicit reverse Hölder inequality for the gradient. The novel feature in the proof is a suitable intrinsic scaling for space-time cylinders combined with reverse Hölder inequalities and a Vitali covering argument within this geometry. The main result holds for the natural range of parameters suggested by other regularity results. Our result applies to general fast diffusion systems and includes both, nonnegative and signed solutions in the case of equations. The methods of proof are purely vectorial in their structure.

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Section: Introduction and Resultsmentioning

confidence: 99%

Higher integrability for the singular porous medium system

Bögelein

Duzaar

Scheven

2019

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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“…The next Lemma provides us with some useful estimates for the quantity b[v, w] that was defined in (3.1). The proof can be found in [2,Lemma 2.3].…”

Section: Preliminariesmentioning

confidence: 99%

An extensive study of the regularity of solutions to doubly singular equations

Vespri

Vestberg

2020

Advances in Calculus of Variations

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In recent years, many papers have been devoted to the regularity of doubly nonlinear singular evolution equations. Many of the proofs are unnecessarily complicated, rely on superfluous assumptions or follow an inappropriate approximation procedure. This makes the theory unclear and quite chaotic to a nonspecialist. The aim of this paper is to fix all the misprints, to follow correct procedures, to exhibit, possibly, the shortest and most elegant proofs and to give a complete and self-contained overview of the theory.

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“…We start by recalling the gluing lemma from the interior case, see [4,Lemma 3.2]. By applying this result to the cylinder Q (θ)…”

Section: 2mentioning

confidence: 99%

Global higher integrability of weak solutions of porous medium systems

Moring¹,

Scheven²,

Schwarzacher³

et al. 2020

Communications on Pure &Amp; Applied Analysis

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

Abstract: In this paper we establish that the gradient of weak solutions to porous medium-type systems admits the self-improving property of higher integrability.

Cited by 41 publications

References 25 publications

Higher integrability for the singular porous medium system

Higher integrability for the singular porous medium system

An extensive study of the regularity of solutions to doubly singular equations

Global higher integrability of weak solutions of porous medium systems

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