On the Global Existence of Weak Solutions for the Navier–Stokes Equations of Compressible Fluid Flows

Abstract

We study the Cauchy problem for the two dimensional Navier–Stokes equations of compressible fluid flows with periodic initial data. We assume that the bulk viscosity coefficient depends on the density of the flow. The global existence of a weak solution with uniform lower and upper bounds on the density, as well as the decay of the solution to an equilibrium state, is proved when the initial datum, $(\rho_0,\,{\bf u}_0)$, does not contain vacuum and belongs to the space $L^\infty\left(\mathbb{T}^2\right) \times \left[W^{1,2}\left(\mathbb{T}^2\right)\right]^2$, where $\mathbb{T}^2 = \mathbb{R}^2/\mathbb{Z}^2$.