Highly oscillatory waves in quasilinear hyperbolic-parabolic coupled equations

Abstract

In this paper, we study the Cauchy problem for a two-speed quasi-linear hyperbolic-parabolic coupled system in several space variables with highly oscillatory initial data and small viscosity. By means of nonlinear geometric optics, we derive the asymptotic expansions of oscillatory waves and deduce that the leading oscillation profiles satisfy quasilinear hyperbolic-parabolic coupled equations with integral terms, from which we obtain that the oscillations of the solutions to the hyperbolic-parabolic equations are propagated along the characteristics of the hyperbolic operators, and partial profiles of oscillations are dissipated by the parabolic effect of the system. Furthermore, by using the energy method in weighted spaces, we rigorously justify the asymptotic expansion and obtain the existence of the highly oscillatory solutions in a time interval independent of the wavelength. Finally, we use this general result to study the behavior of oscillatory waves in the one dimensional compressible viscous flows and in a two-dimensional hyperbolic-parabolic system.