Existence of global solutions to two-dimensional Navier-Stokes equations of a compressible viscous fluid (Q1595719)

The authors consider the two-dimensional Navier-Stokes equations \[ \rho \Biggl(\frac{\partial{ {\mathbf u}}} {\partial t} + ( {\mathbf u} \cdot \nabla) {\mathbf u} + \nabla P = \Delta {\mathbf u} + \nabla ((1+\lambda)\text{div} {\mathbf u}) \Biggr), \frac{\partial{\mathbf u}} {\partial t} + \text{div} (\rho \vec\rho) = 0, \;\lambda = \lambda (\rho),\;P = P (\rho) \] in a class of periodic functions \( {\mathbf u} (x,y)\), \(\rho (x,y)\). The existence and uniqueness of the Cauchy problem is discussed in a Sobolev space. Conditions when a generalized solution is classical are established.