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Gradient regularity for a class of doubly nonlinear parabolic partial differential equations

Michael Strunk

TL;DR

The paper studies local gradient regularity for non-negative weak solutions of the doubly nonlinear parabolic equation $\partial_t u^q - \mathrm{div}\,A(x,t,Du)=0$ under standard $p$-growth structure. The authors prove local Hölder continuity of the spatial gradient in the super-critical fast diffusion regime $0<p-1<q<\frac{n(p-1)}{(n-p)_+}$ by combining a time-insensitive Harnack inequality with Schauder-type gradient estimates for parabolic $p$-Laplacian-type equations, after transforming to $v=u^q$ to obtain a uniformly elliptic coefficient in the transformed equation. They establish an $L^{\infty}$-gradient bound and derive a priori gradient estimates, followed by an approximation argument to handle the case $\mu=0$, ultimately transferring regularity from the simpler model to the general doubly nonlinear setting with $A$ depending on $x$ and $t$. The work extends gradient regularity theory beyond the prototype equation and provides quantitative gradient bounds that depend only on the data, with results localized in space-time. This provides a rigorous framework for gradient regularity in a broad class of doubly nonlinear parabolic equations and lays groundwork for global boundary regularity questions.

Abstract

In this paper, we study the local gradient regularity of non-negative weak solutions to doubly nonlinear parabolic partial differential equations of the type \begin{align*} \partial_t u^q - \mbox{div}\, A(x,t,Du)=0 \qquad\mbox{in $Ω_T$}, \end{align*} with $q>0$, $Ω_T=Ω\times(0,T)\subset\mathbb{R}^{n+1}$ a space-time cylinder, and $A=A(x,t,ξ)$ a vector field satisfying standard $p$-growth conditions. Our main result establishes the local Hölder continuity of the spatial gradient of non-negative weak solutions in the super-critical fast diffusion regime $$0<p-1<q<\frac{n(p-1)}{(n-p)_+}.$$ This result is achieved by utilizing a time-insensitive Harnack inequality and Schauder estimates that are developed for equations of parabolic $p$-Laplacian type. Additionally, we establish a local $L^{\infty}$-bound for the spatial gradient.

Gradient regularity for a class of doubly nonlinear parabolic partial differential equations

TL;DR

The paper studies local gradient regularity for non-negative weak solutions of the doubly nonlinear parabolic equation under standard -growth structure. The authors prove local Hölder continuity of the spatial gradient in the super-critical fast diffusion regime by combining a time-insensitive Harnack inequality with Schauder-type gradient estimates for parabolic -Laplacian-type equations, after transforming to to obtain a uniformly elliptic coefficient in the transformed equation. They establish an -gradient bound and derive a priori gradient estimates, followed by an approximation argument to handle the case , ultimately transferring regularity from the simpler model to the general doubly nonlinear setting with depending on and . The work extends gradient regularity theory beyond the prototype equation and provides quantitative gradient bounds that depend only on the data, with results localized in space-time. This provides a rigorous framework for gradient regularity in a broad class of doubly nonlinear parabolic equations and lays groundwork for global boundary regularity questions.

Abstract

In this paper, we study the local gradient regularity of non-negative weak solutions to doubly nonlinear parabolic partial differential equations of the type \begin{align*} \partial_t u^q - \mbox{div}\, A(x,t,Du)=0 \qquad\mbox{in }, \end{align*} with , a space-time cylinder, and a vector field satisfying standard -growth conditions. Our main result establishes the local Hölder continuity of the spatial gradient of non-negative weak solutions in the super-critical fast diffusion regime This result is achieved by utilizing a time-insensitive Harnack inequality and Schauder estimates that are developed for equations of parabolic -Laplacian type. Additionally, we establish a local -bound for the spatial gradient.
Paper Structure (26 sections, 39 theorems, 312 equations, 1 figure)

This paper contains 26 sections, 39 theorems, 312 equations, 1 figure.

Table of Contents

  1. Introduction and main results
  2. Literature overview
  3. Structure conditions
  4. Main result
  5. Strategy of the proof
  6. Plan of the paper
  7. Preliminaries
  8. Notation
  9. Definition of weak solution
  10. Mollification in time
  11. Auxiliary material
  12. Difference quotients
  13. Structure estimates
  14. Additional material
  15. Gradient bound for parabolic -Laplacian type equations
  16. ...and 11 more sections

Key Result

Theorem 1.1

Let $0<p-1<q<\frac{n(p-1)}{(n-p)_+}$ and $u$ be a non-negative weak solution to pde under assumptions voraussetzungen with $\mu=0$. Then, there exist $\alpha_1=\alpha_1(n,p,q,C_1,C_2,\alpha)\in(0,1)$ and $C=C(n,p,q,C_1,C_2,C_3,\alpha)>1$, such that: for any compact subset $\mathcal{K}_1\Subset \Ome Moreover, we set where $\mathcal{K}_2\Supset\mathcal{K}_1$ is given by Then, there hold the gradi

Figures (1)

  • Figure 1:

Theorems & Definitions (79)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Definition 2.1: Weak solution
  • Definition 2.2: Weak solution to the Cauchy-Dirichlet problem
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • ...and 69 more