A connection between viscous profiles and singular ODEs

TL;DR

This paper explores the relationship between viscous profiles in mixed hyperbolic-parabolic systems, especially the compressible Navier-Stokes equations, and singular ODEs characterized by a division by a potentially zero function.

Contribution

It establishes a connection between viscous profiles and a class of singular ordinary differential equations, providing insights into their structure and behavior.

Findings

Link between viscous profiles and singular ODEs established

Analysis of the singularity when z(V) = 0

Application to the compressible Navier-Stokes equations

Abstract

We deal with the viscous profiles for a class of mixed hyperbolic-parabolic systems. We focus, in particular, on the case of the compressible Navier Stokes equation in one space variable written in Eulerian coordinates. We describe the link between these profiles and a singular ordinary differential equation in the form $$ dV / dt = F(V) / z (V) . $$ Here $V \in R^d$ and the function F takes values into $R^d$ and is smooth. The real valued function z is as well regular: the equation is singular in the sense that z (V) can attain the value 0.