Global classical solutions to the two-dimensional compressible Navier-Stokes equations in ${\mathbb{R}}^2$

Abstract

In this paper, we prove the global well-posedness of the classical solution to the 2D Cauchy problem of the compressible Navier-Stokes equations with arbitrarily large initial data when the shear viscosity $\mu$ is a positive constant and the bulk viscosity $ł(\r)=\r^\b$ with $\b>\frac43$. Here the initial density keeps a non-vacuum states $\bar{\rho }>0$ at far fields and our results generalize the ones by Vaigant-Kazhikhov [41] for the periodic problem and by Jiu-Wang-Xin [26] and Huang-Li [8] for the Cauchy problem with vacuum states $\bar{\rho }=0$ at far fields. It shows that the solution will not develop the vacuum states in any finite time provided the initial density is uniformly away from vacuum. And the results also hold true when the initial data contains vacuum states in a subset of ${\mathbb{R}}^2$ and the natural compatibility conditions are satisfied. Some new weighted estimates are obtained to establish the upper bound of the density.

30 pages. arXiv admin note: substantial text overlap with arXiv:1207.5874, arXiv:1202.1382

References in corpus (4)

• Vanishing of vacuum states and blow-up phenomena of the compressible Navier-Stokes equations
• Global Well-Posedness of Classical Solutions to the Cauchy problem of Two-Dimensional Baratropic Compressible Navier-Stokes System with Vacuum and Large Initial Data
• Existence and Blowup Behavior of Global Strong Solutions to the Two-Dimensional Baratropic Compressible Navier-Stokes System with Vacuum and Large Initial Data
• On blowup of classical solutions to the compressible Navier-Stokes equations

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• Global classical solution of the Cauchy problem to 1D compressible Navier-Stokes equations with large initial data