Invariant manifolds for a singular ordinary differential equation

TL;DR

This paper investigates invariant manifolds near singular points in a class of singular ODEs, extending classical concepts like center and stable manifolds, with applications to viscous profiles in hyperbolic-parabolic systems including Navier-Stokes.

Contribution

It extends the theory of invariant manifolds to singular ODEs where the denominator function can vanish, providing new tools for analyzing solutions near singularities.

Findings

Existence of locally invariant manifolds near singular points.

Extension of center and stable manifold concepts to singular ODEs.

Application to viscous profiles in hyperbolic-parabolic systems, including Navier-Stokes.

Abstract

We study the singular ordinary differential equation $$ \frac{d U}{d t} = f (U) / z (U) + g (U), $$ where $U \in R^N$, the functions $f \in R^N $ and $g \in R^N $ are of class $C^2$ and $z $ is a real valued $C^2$ function. The equation is singular in the sense that $z (U)$ can attain the value 0. We focus on the solutions of the singular ODE that belong to a small neighborhood of a point $\bar U$ such that $f (\bar U) = g (\bar U) = \vec 0$, $z (\bar U) =0$. We investigate the existence of manifolds that are locally invariant for the singular ODE and that contain orbits with a suitable prescribed asymptotic behaviour. Under suitable hypotheses on the set $\{U: z (U) = 0 \}$, we extend to the case of the singular ODE the definitions of center manifold, center stable manifold and of uniformly stable manifold. An application of our analysis concerns the study of the viscous profiles with small total variation for a class of mixed hyperbolic-parabolic systems in one space variable. Such a class includes the compressible Navier Stokes equation.