Existence of traveling waves in van der Waals fluids with viscosity and capillarity effects

We establish the existence of traveling waves of non-isentropic van der Waals fluids with viscosity and capillarity effects. The method developed the one for simpler models. The nonconvex equation of state of the fluid causes much difficulty in evaluating the related quantities, and so the argument and the analysis are much more involved than the convex equation of state. The point is to estimate the pressure along the Hugoniot curves such that a Lyapunov function can be defined in an appropriate way.