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Local boundedness and higher integrability for the sub-critical singular porous medium system

Verena Bögelein, Frank Duzaar, Ugo Gianazza, Naian Liao

TL;DR

This work resolves the long-standing question of higher integrability for the gradient of weak solutions to porous medium-type systems in the critical and sub-critical fast-diffusion range, proving Du^m∈L^{2+2ε_o}_{loc} under N≥3, 0<m≤m_c with suitable integrability of F and a local L^r condition on u. The authors develop non-uniform intrinsic cylinders whose space-time scaling depends on the mean of |u|^r, combine Moser–De Giorgi iterations to obtain local boundedness, and establish Sobolev–Poincaré and reverse Hölder inequalities to achieve a quantitative higher-integrability result with scaling deficit d=2r/λ_r. A Vitali-covering argument and a stopping-time technique culminate in a robust gradient bound on Du^m that adapts to the nonlinearity and the inhomogeneous scaling, with explicit data-dependence and sharpness considerations. The results extend the theory to systems and to more general growth conditions on A, providing a comprehensive completion of the higher-integrability program for porous medium-type equations in the fast-diffusion regime and offering tools with potential applications to related nonlinear parabolic systems.

Abstract

The gradient of weak solutions to porous medium-type equations or systems possesses a higher integrability than the one assumed in the pure notion of a solution. We settle the critical and sub-critical, singular case and complete the program.

Local boundedness and higher integrability for the sub-critical singular porous medium system

TL;DR

This work resolves the long-standing question of higher integrability for the gradient of weak solutions to porous medium-type systems in the critical and sub-critical fast-diffusion range, proving Du^m∈L^{2+2ε_o}_{loc} under N≥3, 0<m≤m_c with suitable integrability of F and a local L^r condition on u. The authors develop non-uniform intrinsic cylinders whose space-time scaling depends on the mean of |u|^r, combine Moser–De Giorgi iterations to obtain local boundedness, and establish Sobolev–Poincaré and reverse Hölder inequalities to achieve a quantitative higher-integrability result with scaling deficit d=2r/λ_r. A Vitali-covering argument and a stopping-time technique culminate in a robust gradient bound on Du^m that adapts to the nonlinearity and the inhomogeneous scaling, with explicit data-dependence and sharpness considerations. The results extend the theory to systems and to more general growth conditions on A, providing a comprehensive completion of the higher-integrability program for porous medium-type equations in the fast-diffusion regime and offering tools with potential applications to related nonlinear parabolic systems.

Abstract

The gradient of weak solutions to porous medium-type equations or systems possesses a higher integrability than the one assumed in the pure notion of a solution. We settle the critical and sub-critical, singular case and complete the program.
Paper Structure (24 sections, 20 theorems, 412 equations)

This paper contains 24 sections, 20 theorems, 412 equations.

Table of Contents

  1. Introduction and main results
  2. Models
  3. Novelty and Significance
  4. Sharpness of the result
  5. Main technical tools
  6. Structure of the paper
  7. Notation and definition
  8. Notation
  9. Notion of weak solution
  10. Auxiliary material
  11. Local boundedness of weak solutions
  12. Energy bounds
  13. Sobolev-Poincaré-type inequality
  14. Reverse Hölder inequality
  15. Proof of the higher integrability
  16. ...and 9 more sections

Key Result

Theorem 1.1

Assume that $N\ge3$, and $r>0$ that satisfies Then, there exists $\varepsilon_o{=}\varepsilon_o(N,m,\nu,L, p,r){\in} (0,1]$ such that whenever $F\in L^{2p}_{\rm loc}(\Omega_T,\mathbb{R}^{kN})$, $u$ is a weak solution to the porous medium-type system por-med-eq in the sense of Definition def:weak_solution under the assumptions growth, and $u\in Moreover, for every $\varepsilon\in(0,\varepsilon_o

Theorems & Definitions (38)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Corollary 1.5
  • Definition 2.1
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • ...and 28 more