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

作者：Guochun Wu, Lei Yao, Yinghui Zhang · 发表于：SIAM Journal on Mathematical Analysis · 年份：2023 · DOI：10.1137/22m1478859 · 被引用次数：7 · 研究领域：Navier-Stokes equation solutions、Advanced Mathematical Physics Problems、Computational Fluid Dynamics and Aerodynamics

.We investigate global stability and nonvanishing vacuum states of large solutions to the compressible Navier–Stokes equations on the torus \(\mathbb{T}^3\) , and the main purpose of this work is threefold. First, under the assumption that the density \(\rho ({\mathbf{x}}, t)\) verifies \(\sup_{t\geq 0}\|\rho ( t)\|_{L^\infty }\leq M\) , it is shown that the solutions converge to an equilibrium state exponentially in the \(L^2\) -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 \(\rho_0({\mathbf{x}})\) satisfies \(\inf_{{\mathbf{x}}\in \mathbb{T}^3}\rho_0({\mathbf{x}})\geq c_0\gt 0\) . 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 \(\mu (\rho )=\rho^\alpha\) with \(\alpha \gt 1/2\) . This phenomenon imp...