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Quantitative stability for quasilinear parabolic equations

Tapio Kurkinen, Qing Liu

TL;DR

The paper develops a unified, quantitative stability framework for a broad class of quasilinear parabolic equations with potential gradient-vanishing singularities, using ${\mathcal F}$-solutions and a doubling-variable, Crandall–Ishii-based approach to obtain explicit convergence rates as perturbations vanish. It applies to both normalized and variational $p$-Laplace parabolic equations and to regularized approximations, yielding rates that depend on Hölder regularity via $\nu=\frac{\alpha\theta}{1+(1-\theta)\max\{\beta,0\}}$. The results extend to elliptic analogs and connect to vanishing-viscosity limits for Hamilton–Jacobi equations, highlighting explicit, implementable stability estimates for nonlinear parabolic models. By providing concrete rate formulas in multiple $p$-Laplace contexts and for various regularizations, the work offers practical guidance for numerical approximations and convergence analyses of degenerate or singular parabolic problems. Overall, it fills a gap in the literature by turning qualitative stability into actionable, quantitative stability with verifiable assumptions and clear dependence on regularity and problem parameters.

Abstract

We examine the stability of a class of quasilinear parabolic partial differential equations under perturbations. We are interested in the behavior of viscosity solutions as the perturbation parameter vanishes and establish explicit convergence rates by adapting standard comparison arguments. Despite the possible singular or degenerate nature of the parabolic operator, our framework covers, in particular, both the normalized and the variational $p$-parabolic equations, providing quantitative estimates for perturbations of the exponent $p$ and limits arising from regularized approximations.

Quantitative stability for quasilinear parabolic equations

TL;DR

The paper develops a unified, quantitative stability framework for a broad class of quasilinear parabolic equations with potential gradient-vanishing singularities, using -solutions and a doubling-variable, Crandall–Ishii-based approach to obtain explicit convergence rates as perturbations vanish. It applies to both normalized and variational -Laplace parabolic equations and to regularized approximations, yielding rates that depend on Hölder regularity via . The results extend to elliptic analogs and connect to vanishing-viscosity limits for Hamilton–Jacobi equations, highlighting explicit, implementable stability estimates for nonlinear parabolic models. By providing concrete rate formulas in multiple -Laplace contexts and for various regularizations, the work offers practical guidance for numerical approximations and convergence analyses of degenerate or singular parabolic problems. Overall, it fills a gap in the literature by turning qualitative stability into actionable, quantitative stability with verifiable assumptions and clear dependence on regularity and problem parameters.

Abstract

We examine the stability of a class of quasilinear parabolic partial differential equations under perturbations. We are interested in the behavior of viscosity solutions as the perturbation parameter vanishes and establish explicit convergence rates by adapting standard comparison arguments. Despite the possible singular or degenerate nature of the parabolic operator, our framework covers, in particular, both the normalized and the variational -parabolic equations, providing quantitative estimates for perturbations of the exponent and limits arising from regularized approximations.
Paper Structure (12 sections, 8 theorems, 174 equations)

This paper contains 12 sections, 8 theorems, 174 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Main results
  4. Applications
  5. Preliminaries
  6. Quantitative stability
  7. Applications and examples
  8. The normalized $p$-Laplace type equations
  9. The variational $p$-Laplace type equations
  10. General $p$-Laplace type equations
  11. Regularization for generalized $p$-Laplace equations
  12. Regularization for biased infinity-Laplace equation

Key Result

Theorem 1.1

Let $T>0$ and $\Omega$ be a bounded domain in ${\mathbb R}^n$. For $\varepsilon\geq 0$, assume that $A_\varepsilon: {\mathbb R}^n_0\to {\mathcal{S}}_+^n$ and ${H_\varepsilon: \Omega_T\times {\mathbb R}^n\to {\mathbb R}}$ are continuous functions satisfying (A1)(A2) and (C1)(C2). Let $g_\varepsilon\i for all $(x_1, t_1), (x_2, t_2)\in \overline{\Omega}\times [0, T)$ and $\varepsilon\geq 0$ small. T

Theorems & Definitions (22)

  • Theorem 1.1: Quantitative stability for parabolic equations
  • Definition 2.1: Compatible test functions
  • Definition 2.2
  • Definition 2.3
  • proof : Proof of Theorem \ref{['thm:parabolic']}
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Theorem 3.4
  • Remark 3.5
  • ...and 12 more