Time asymptotic behavior of non-equilibrium flows in one space dimension

Abstract

We study long time behavior of polyatomic gas flows in both translational and vibrational non-equilibrium. The author previously established global existence of solution and obtained optimal $L^2$ time-decay rates for the solution towards an equilibrium state for the Cauchy problem. The current paper is a continuation in studying the solution behavior. An asymptotic solution is constructed explicitly using a heat kernel along the particle path and two Burgers kernels along the equilibrium acoustic directions. Convergence of the exact solution to the asymptotic solution is studied in a pointwise sense in both space and time to give a complete picture of wave propagation. The study lays a foundation for a future work on solution behavior around a shock wave, a mechanism that induces Richtmyer–Meshkov instability in mixing problems .