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Higher integrability for the singular porous medium system

Verena Bögelein, Frank Duzaar, Christoph Scheven

Abstract

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.

Higher integrability for the singular porous medium system

Abstract

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.

Paper Structure

This paper contains 16 sections, 12 theorems, 145 equations.

Table of Contents

  1. Introduction and results
  2. Notation and main result
  3. Notations
  4. General Setting and Results
  5. Auxiliary Material
  6. Energy bounds
  7. Sobolev-Poincaré type inequality
  8. Reverse Hölder inequality
  9. Proof of the higher integrability
  10. Construction of a non-uniform system of cylinders
  11. Covering property
  12. Stopping time argument
  13. A Reverse Hölder Inequality
  14. Estimate on super-level sets
  15. Proof of the gradient estimate
  16. ...and 1 more sections

Key Result

Theorem \oldthetheorem

Assume that and $\sigma>2$. Then, there exists $\varepsilon_o=\varepsilon_o(n,m,\nu,L)\in (0,1]$ such that whenever $F\in L^\sigma(\Omega_T,\mathbb{R}^{Nn})$ and $u$ is a weak solution of Equation por-med-eq in the sense of Definition def:weak_solution under the assumptions growth, then with $\varepsilon_1:=\mi Moreover, for every $\varepsilon\in(0,\varepsilon_1]$ and every cylinder $Q_{2R}(z_o)\

Theorems & Definitions (19)

  • Definition \oldthetheorem
  • Theorem \oldthetheorem
  • Remark \oldthetheorem
  • Remark \oldthetheorem
  • Corollary \oldthetheorem
  • Lemma \oldthetheorem
  • Lemma \oldthetheorem
  • Lemma \oldthetheorem
  • Lemma \oldthetheorem
  • Lemma \oldthetheorem
  • ...and 9 more