GLOBAL STABILITY AND NONVANISHING VACUUM STATES OF 3D COMPRESSIBLE NAVIER-STOKES EQUATIONS

We investigate global stability and nonvanishing vacuum states of large solutions to the compressible Navier-Stokes equations on the torus T3, and the main purpose of this work is threefold. First, under the assumption that the density p(x,t) verifies supt\geq 0 \|p(t)\|L\infty \leq M, it is shown that the solutions converge to an equilibrium state exponentially in the L2-norm. In contrast to previous related works where the density has uniform positive lower and upper bounds, this gives the first stability result for large strong solutions of the three-dimensional compressible Navier-Stokes equations in the presence of vacuum. Second, by employing some new thoughts, we also show that the density converges to its equilibrium state exponentially in the L\infty-norm if additionally the initial density p0(x) satisfies inf\bfx \inT3 p0(x) \geq c0 > 0. Finally, we prove that the vacuum state will persist for any time provided that the initial density contains vacuum, which is different from the previous work of [H. L. Li, J. Li, and Z. P. Xin, Comm. Math. Phys., 281 (2008), pp. 401--444], where the authors showed that any vacuum state must vanish within finite time for the free boundary problem of the one-dimensional compressible Navier-Stokes equations with density-dependent viscosity \mu(p) = p\alpha with \alpha > 1/2. This phenomenon implies the different behaviors for Navier--Stokes equations with different types of viscous effects, namely, degenerate or not.