With External Pressure Stability Of The Weak Solution Of The Navier-stokes Equations

Recently, compressible Navier-Stokes equation with density-dependent viscosity has been derived in the analysis of dynamical behavior of shallow water[7], which can be also deduced from fluid-dynamical asympotical approximation of Boltzmann equation[3] (under isentropic approximation). It has been extensively studied in recent years, since the behavior of solution to Navier-Stokes equation is very complicated on the boundary of vacuum. The complication in this ease has been verified by some related studies (see the introduction in Chapter 1). In this paper, we are concerned with the initial value problem on spatial periodic domain where γ > 1, the initial values are given byand there exists δ > 0 (small enough) such thatWhen proved the stability of the weak solution of initial value problem (0.1)-(0.3) on spatial periodic domain Ω = T~N. This result improves the previous results on compressible Navier-Stokes with external force with the restriction of γ>N/2 (the reader can refer to [2] and references therein), and extends the stability result on weak solutions to compressible Navier-Stokes [1] to the case with exteral force involved.