Global well-posedness of the Cauchy problem of two-dimensional compressible Navier-Stokes equations in weighted spaces

Abstract

In this paper, we study the global well-posedness of classical solution to 2D Cauchy problem of the compressible Navier-Stokes equations with large initial data and vacuum. It is proved that if the shear viscosity $\mu$ is a positive constant and the bulk viscosity $\lambda$ is the power function of the density, that is, $\lambda (\rho )={\rho }^{\beta }$ with $\beta >3$, then the 2D Cauchy problem of the compressible Navier-Stokes equations on the whole space ${\mathbf{R}}^2$ admit a unique global classical solution $(\rho ,u)$ which may contain vacuums in an open set of ${\mathbf{R}}^2$. Note that the initial data can be arbitrarily large to contain vacuum states. Various weighted estimates of the density and velocity are obtained in this paper and these self-contained estimates reflect the fact that the weighted density and weighted velocity propagate along with the flow.

47 pages. arXiv admin note: substantial text overlap with arXiv:1202.1382

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