Global Stability and Non-Vanishing Vacuum States of 3D Compressible Navier-Stokes Equations

We investigate the global stability and non-vanishing vacuum states of large solutions to the compressible Navier-Stokes equations on the torus $\mathbb{T}^3$, and the main novelty of this work is three-fold: First, under the assumption that the density $\rho({\bf{x}}, t)$ verifies $\sup_{t\geq 0}\|\rho(t)\|_{L^\infty}\leq M$, it is shown that the solutions converge to equilibrium state exponentially in $L^2$-norm. Second, by employing some new thoughts, we also show that the density converges to its equilibrium state exponentially in $L^\infty$-norm if additionally the initial density $\rho_0({\bf{x}})$ satisfies $\inf_{{\bf{x}}\in\mathbb{T}^3}\rho_0({\bf{x}})\geq c_0>0$. Finally, we prove that the vacuum states will not vanish for any time provided that the vacuum states are present initially. This phenomenon is totally new and somewhat surprising, and particularly is in contrast to the previous work of [H. L. Li et al., Commun. Math. Phys., 281 (2008), 401-444], where the authors showed that the vacuum states must vanish within finite time for the 1D compressible Navier-Stokes equations with density-dependent viscosity $\mu(\rho)=\rho^\alpha$ with $\alpha>1/2$.Comment: 17 page

On weak (measure-valued)-strong uniqueness for compressible Navier-stokes system with non-monotone pressure law

In this paper our goal is to define a renormalized dissipative measure-valued (rDMV) solution of compressible Navier–Stokes system for fluids with non-monotone pressure–density relation. We prove existence of rDMV solutions and establish a suitable relative energy inequality. Moreover we obtain the weak (measure-valued)–strong uniqueness property of this rDMV solution with the help of relative energy inequality