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

TL;DR

This paper proves the global stability of solutions to 3D compressible Navier-Stokes equations on a torus, showing exponential convergence to equilibrium and the non-vanishing of vacuum states over time, which contrasts with previous finite-time vacuum vanishing results.

Contribution

It establishes the exponential convergence of solutions to equilibrium and demonstrates that vacuum states persist indefinitely under certain conditions, introducing new insights into vacuum behavior in 3D compressible flows.

Findings

Solutions converge exponentially in L^2-norm to equilibrium.

Density converges exponentially in L^∞-norm if initial density is bounded below.

Vacuum states do not vanish over time if initially present.

Abstract

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 $\alpha>1/2$.