The Asymptotic Stability Of The Model Navier-Stokes Equations With Viscous Sparse Wave Solutions

We investigate the model Navier-Stokes equations . We construct the solution of rarefaction waves and obtain its asymptotic stability for the Cauchy problem by energy estimations. There are three chapters in this paper. We first give a simple introduction of the history of the asymptotic behavior of the solution of viscous rarefaction waves to the conservation laws with viscosity . In the second chapter, we describe the results of others and our main results. Our main result says that if the initial data is close to a constant state and its values at ±? oo lie on the Kth rarefaction curve for corresponding Euler equations, then the solution tends as t ?? to the viscous rarefaction wave determined by these states. Finally we prove some priori estimates by energy method and then give the proof of our main results by the continuous induction.