Asymptotic Stability Of Shock Wave Solutions Of A One-Dimensional Model System For Compressible Viscous Fluids

In this paper, we investigate the initial value and the boundary value problem of a one-dimensional model system for compressible viscous dynamic fluids in the Eulerian coordinates. A one-dimensional model system for compressible viscous fluids is quasilinear hyperbolic partial differential equations. In addition to continuous solutions, its solution also could be intermittent; intermittent solutions can be generated or disappear. Continuous flow can be described as simple wave, such as compression wave and rarefaction wave; intermittent solutions perform for the shock wave of moving fluids. Shock wave is not only a physical phenomena, but also mathematical characteristics which need to be concerned about when solving equations for fluid dynamics. In this paper, the model system for compressible viscous fluids is discussed under isentropic situation. By constructing superposition of the shock wave and using compressed mapping theory, we proof the local existence and uniqueness of the shock wave. The prior estimate is proved by elementary energy method, and unique global existence and asymptotic property of the solution are established. The results of numerical calculation by difference method proved that superposition of the shock waves is the solution of viscous fluids. The main results and methods are as follows:1, Construct superposition of the shock waves: Using Lagrangian coordinates transform and Rankine-Hugoniot conditions, we get viscous shock curve S1 and S2 in the Lagrangian coordinates, which are used to construct superposition of the shock waves. The displacement of the shock wave is calculated by using the initial value; superposition of the shock waves satisfies the initial boundary conditions.2, Unique local existence of the solution: Superposition of shock waves structured by iterative methods come to be a Cauchy sequence, then the local existence of solutions of compressible fluids are proved by the compressed mapping principle; using energy estimate and all kinds of inequality, the local uniqueness of solutions of compressible fluids are proved.3, Unique global existence and asymptotic property of the solution: The proof of the priori estimate is finished through estimate inequality of the corresponding variables by elementary energy method. Unique global existence and asymptotic property of the solution are based on priori estimates and prolongation theorem.