Long-time behavior of weak solutions for compressible Navier-Stokes equations with degenerate viscosity

The long-time regularity and asymptotic of weak solutions are studied for compressible Navier-Stokes equations with degenerate viscosity in a bounded periodic domain in two and three dimensions. It is shown that the density keeps strictly positive from below and above after a finite period of time. Moreover, higher velocity regularity is obtained via a parabolic type iteration technique. Since then the weak solution conserves its energy equality, and decays exponentially to the equilibrium in $L^{2}$-norm as time goes to infinity. In addition, assume that the initial momentum is zero, the exponential decay rate is derived for the derivative functions, and the weak solution becomes a strong one in two dimensional space.