Nonlinear Stability of the Monotone Traveling Wave for the Isothermal Fluid Equations with Viscous and Capillary Terms

Abstract

We prove the existence of the monotone traveling wave for the isothermal fluid equations with viscous and capillary terms by the planar dynamical system method. We obtain that the monotone traveling wave is asymptotically stable under the suitable perturbation. In the process of establishing the uniform a priori estimate, we dispose the capillary term reasonably according to the feature of the equations, and find the appropriate weighted function to overcome the difficulty caused by the non-convex pressure function.