Stability properties and dynamics of solutions to viscous conservation laws with mean curvature operator

TL;DR

This paper investigates the long-term behavior of solutions to a scalar conservation law with nonlinear diffusion, highlighting metastability phenomena and providing numerical evidence for the stability and layered structures of solutions.

Contribution

It introduces the analysis of metastability in viscous conservation laws with mean curvature operators and demonstrates the existence, stability, and layered structures of solutions.

Findings

Existence of a unique stationary solution

Metastability phenomena observed in solutions

Numerical simulations confirm theoretical results

Abstract

In this paper we study the long time dynamics of the solutions to the initial-boundary value problem for a scalar conservation law with a saturating nonlinear diffusion. After discussing the existence of a unique stationary solution and its asymptotic stability, we focus our attention on the phenomenon of 'metastability', whereby the time-dependent solution develops into a layered function in a relatively short time, and subsequently approaches a steady state in a very long time interval. Numerical simulations illustrate the results.