Asymptotic Stability Of Solution To Viscous Conservation Laws

Navier-Stokes Equations is the basic model in fluid dynamics. When viscosity tends to 0, the solution to Navier-Stokes Equations tends to that of corresponding Euler Equations ([28]). By a coordinate transformation, the discussion on the behavior of solutions as viscosity vanishes can be converted to that of the large time behavior of the solutions to Navier-Stokes Equations with fixed viscosity. Therefore, the large time behavior of the global solution to viscous conservation laws has become one of the most important topics in fluid dynamics. In the past decades, there have been a lot of works on the asymptotic stability of viscous wave (rarefaction wave, viscous shock profile, stationary wave) to the Cauchy problems and initial boundary value problems of viscous conservation laws([l-23,25-28]).In this paper, we consider Cauchy problems and initial-boundary value problems of one-dimensional isentropic Euler system and 2-dimensional steady isentropic irrotational planar flow with fixed viscosity.The paper is organized as follows:In chapter 1, we briefly introduce the progress in history and some recent results in this field, and give the background of the problems. A more detailed arrangement of the paper, the detailed results and methods in proofs are also included in this Chapter.In Chapter 2, we study asymptotic stability of the viscous shock to the initial-boundary value problem of one-dimensional isentropic Euler system with viscosity, which can be regarded as a perturbation of impermeable problem. If there exists a viscous shock profile to the corresponding Cauchy problem of viscous conservation laws, we prove that the viscous shock is asymptotically stable under some assumption.In Chapter 3, we consider two Cauchy problems of the 2-dimensional steady isentropic irrotational planar flow with artificial viscosity. If the initial data are suitably close to a of the corresponding inviscid hyperbolic system admits a weak rarefaction wave, then the solutions to the two problems of the viscous conservation laws tend to this weak rarefaction wave as t tends to infinity.Based on the results in Chapter 3, Chapter 4 is devoted to the asymptotic stability of the stationary wave to the initial-boundary problem of the 2-dimensional steady isentropic irrotational planar flow with artificial viscosity.