Notes on the study of the viscous approximation of hyperbolic problems via ODE analysis

TL;DR

This paper explores how ordinary differential equation analysis, especially invariant manifolds, can be applied to understand the viscous approximation of hyperbolic conservation laws, including extensions to singular ODEs related to Navier-Stokes.

Contribution

It introduces a framework for applying invariant manifold theory to viscous approximations of hyperbolic problems and extends these concepts to certain singular ODEs in fluid dynamics.

Findings

Analysis of invariant manifolds aids in understanding viscous approximations.

Extension of manifold concepts to singular ODEs in Navier-Stokes.

Potential new methods for studying hyperbolic PDEs via ODE techniques.

Abstract

These notes describe some applications of the analysis of ordinary differential equations to the study of the viscous approximation of conservation laws in one space dimension. The exposition mostly focuses on the analysis of invariant manifolds like the center manifold and the stable manifold. The last section addresses a more specific issue and describes a possible way of extending the notions of center and stable manifold to some classes of singular ordinary differential equations arising in the study of the Navier-Stokes equation in one space variable.