Global Behavior of Spherically Symmetric Navier-Stokes Equations with Density-Dependent Viscosity

In this paper, we study a free boundary problem for compressible spherically symmetric Navier-Stokes equations without a solid core. Under certain assumptions imposed on the initial data, we obtain the global existence and uniqueness of the weak solution, give some uniform bounds (with respect to time) of the solution and show that it converges to a stationary one as time tends to infinity. Moreover, we obtain the stabilization rate estimates of exponential type in $L^\infty$-norm and weighted $H^1$-norm of the solution by constructing some Lyapunov functionals. The results show that such system is stable under the small perturbations, and could be applied to the astrophysics.Comment: 38 page

Decay Estimates for Isentropic Compressible Navier-Stokes Equations in Bounded Domain

In this paper, under the hypothesis that $\rho$ is upper bounded, we construct a Lyapunov functional for the multidimensional isentropic compressible Navier-Stokes equations and show that the weak solutions decay exponentially to the equilibrium state in $L^2$ norm. This can be regarded as a generalization of Matsumura and Nishida's results in 1982, since our analysis is done in the framework of Lions 1998 and Feireisl et al. 2001, the higher regularity of $(\rho, u)$ and the uniformly positive lower bound of $\rho$ are not necessary in our analysis and vacuum may be admitted. Indeed, the upper bound of the density $\rho$ plays the essential role in our proof.Comment: 9 page