Asymptotic stability of strong rarefaction waves for the compressible fluid models of Korteweg type

Abstract

This paper is concerned with the time-asymptotic behavior toward strong rarefaction waves of solutions to one-dimension compressible fluid models of Korteweg type, which governs the motions of the compressible fluids with internal capillarity. Assume that the corresponding Riemann problem to the compressible Euler system can be solved by rarefaction waves (VR,UR)(t,x). If the initial data is a small perturbation of an approximate rarefaction wave for (VR,UR)(t,x), we show that the corresponding Cauchy problem admits a unique global smooth solution which tends to (VR,UR)(t,x) time asymptotically. Since we do not require the strength of the rarefaction waves to be small, this result gives the nonlinear stability of strong rarefaction waves for the one-dimensional compressible fluid models of Korteweg type. The analysis is based on the elementary L2 energy method together with continuation argument.