Global Stability and Nonvanishing Vacuum States of 3D Compressible Navier–Stokes Equations

Abstract

We investigate global stability and nonvanishing vacuum states of large solutions to the compressible Navier–Stokes equations on the torus , and the main purpose of this work is threefold. First, under the assumption that the density verifies , it is shown that the solutions converge to an equilibrium state exponentially in the -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 -norm if additionally the initial density satisfies . 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 with . This phenomenon implies the different behaviors for Navier–Stokes equations with different types of viscous effects, namely, degenerate or not.