Invariant manifolds for a singular ordinary differential equation

We study the singular ordinary differential equationequation(0.1)dUdt=1ζ(U)ϕs(U)+ϕns(U), where U∈RNU∈RN, the functions ϕs∈RNϕs∈RN and ϕns∈RNϕns∈RN are of class C2C2 and ζ   is a real valued C2C2 function. The equation is singular because ζ(U)ζ(U) can attain the value 0. We focus on the solutions of (0.1) that belong to a small neighborhood of a point U¯ such that ϕs(U¯)=ϕns(U¯)=0→ and ζ(U¯)=0. We investigate the existence of manifolds that are locally invariant for (0.1) and that contain orbits with a prescribed asymptotic behavior. Under suitable hypotheses on the set {U:ζ(U)=0}{U:ζ(U)=0}, we extend to the case of the singular ODE (0.1) the definitions of center manifold, center-stable manifold and of uniformly stable manifold. We prove that the solutions of (0.1) lying on each of these manifolds are regular: this is not trivial since we provide examples showing that, in general, a solution of (0.1) is not continuously differentiable. Finally, we show a decomposition result for a center-stable manifold and for the 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.