Metastability for nonlinear parabolic equations with application to scalar viscous conservation laws

TL;DR

This paper develops a general framework for understanding metastability in one-dimensional parabolic PDEs, focusing on scalar viscous conservation laws, by analyzing approximate steady states and spectral properties.

Contribution

It introduces a versatile approach to metastability in PDEs using approximate steady states and spectral analysis, applicable to scalar viscous conservation laws with boundary conditions.

Findings

Established a reduced system for metastable dynamics involving an ODE and PDE.

Proved a general result for the quasi-linearized system ignoring nonlinear terms.

Applied the framework successfully to scalar viscous conservation laws with Dirichlet boundary conditions.

Abstract

The aim article is to contribute to the definition of a versatile language for metastability in the context of partial differential equations of evolutive type. A general framework suited for parabolic equations in one dimensional bounded domains is proposed, based on choosing a family of approximate steady states, and on the spectral properties of the linearized operators at such states. The slow motion for solutions belonging to a cylindrical neighborhood of the family of approximate steady states is analyzed by means of a system of an ODE for the parameter $\xi$ that describes the family, coupled with a PDE describing the evolution of the perturbation $v$. We state and prove a general result concerning the reduced system for the couple $(\xi,v)$, called quasi-linearized system, obtained by disregarding the nonlinear term in $v$, and we show how such approach suits to the prototypical example of scalar viscous conservation laws with Dirichlet boundary condition in a bounded one-dimensional interval with convex flux.