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On equivalence of entropy and viscosity solutions to degenerate parabolic equations and applications

Hiroyoshi Mitake, Hiroshi Watanabe

TL;DR

This work establishes a precise equivalence between entropy solutions of anisotropic, degenerate parabolic-hyperbolic equations (CL) and viscosity solutions of degenerate quasilinear Hamilton-Jacobi equations (HJ), showing that, under a framework of structural assumptions, the derivative relation $u=\partial_x v$ pairs these two weak-solution notions. The main result proves two-way implications: from a Lipschitz viscosity solution $v$ of (HJ) one obtains the entropy solution $u=\partial_x v$ of (CL), and from an entropy solution $u$ of (CL) there exists a time-dependent additive function $\widehat{C}(t)$ such that $v(x,t)=\int_0^x u(y,t)\,dy + \widehat{C}(t)$ is a viscosity solution to (HJ). The paper also derives large-time behavior in the periodic setting, leveraging Panov-type results for (CL) and ergodic problem theory for (HJ), thereby describing the asymptotic profiles of solutions and their primitives. Overall, this work unifies entropy- and viscosity-based frameworks for a broad class of degenerate parabolic equations and provides concrete long-time asymptotics in periodic domains, enriching the theory and potential applications to nonlinear diffusion and Hamilton-Jacobi-type models.

Abstract

Here, we consider anisotropic degenerate parabolic-hyperbolic equations and degenerate quasilinear Hamilton-Jacobi equations. We prove the equivalence of two notions of entropy and viscosity solutions of two equations, and apply it to obtain a large-time behavior of viscosity solutions to quasilinear Hamilton-Jacobi equations, and entropy solutions to degenerate parabolic-hyperbolic equations in a periodic setting.

On equivalence of entropy and viscosity solutions to degenerate parabolic equations and applications

TL;DR

This work establishes a precise equivalence between entropy solutions of anisotropic, degenerate parabolic-hyperbolic equations (CL) and viscosity solutions of degenerate quasilinear Hamilton-Jacobi equations (HJ), showing that, under a framework of structural assumptions, the derivative relation pairs these two weak-solution notions. The main result proves two-way implications: from a Lipschitz viscosity solution of (HJ) one obtains the entropy solution of (CL), and from an entropy solution of (CL) there exists a time-dependent additive function such that is a viscosity solution to (HJ). The paper also derives large-time behavior in the periodic setting, leveraging Panov-type results for (CL) and ergodic problem theory for (HJ), thereby describing the asymptotic profiles of solutions and their primitives. Overall, this work unifies entropy- and viscosity-based frameworks for a broad class of degenerate parabolic equations and provides concrete long-time asymptotics in periodic domains, enriching the theory and potential applications to nonlinear diffusion and Hamilton-Jacobi-type models.

Abstract

Here, we consider anisotropic degenerate parabolic-hyperbolic equations and degenerate quasilinear Hamilton-Jacobi equations. We prove the equivalence of two notions of entropy and viscosity solutions of two equations, and apply it to obtain a large-time behavior of viscosity solutions to quasilinear Hamilton-Jacobi equations, and entropy solutions to degenerate parabolic-hyperbolic equations in a periodic setting.
Paper Structure (6 sections, 9 theorems, 63 equations)

This paper contains 6 sections, 9 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Equivalence
  4. Applications
  5. Large-time behavior for \ref{['HJ']}
  6. Large-time behavior for \ref{['CL']}

Key Result

Theorem 1.1

Assume that (A1)--(A5) hold. Let $v \in {\rm Lip\,}(\mathbb{R} \times [0,\infty))$ be the unique viscosity solution to HJ. Then, $u = \partial_{x}v\in C_{{\rm loc}}([0,\infty) ; L^{1}(\mathbb{R})) \cap L^{\infty}(\mathbb{R} \times (0,\infty))$ is the unique entropy solution to CL. Conversely, let $u

Theorems & Definitions (20)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1: Entropy solution
  • Definition 2.2: Viscosity sub/super solution
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • ...and 10 more