A connection between viscous profiles and singular ODEs

We deal with the viscous profiles for a class of mixed hyperbolic-parabolic systems in one space dimension. 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 = 1/ζ(V) F(V). (1) Here V ∈ Rd and the function F takes values into Rd and is smooth. The real valued function ζ is as well regular: the equation is singular in the sense that ζ(V) can attain the value 0.