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
μ
is a positive constant and the bulk viscosity
λ
is the power function of the density, that is,
λ(ρ)=
ρ
β
with
β>3
, then the 2D Cauchy problem of the compressible Navier-Stokes equations on the whole space
R
2
admit a unique global classical solution
(ρ,u)
which may contain vacuums in an open set of
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.