  The BGK model kinetic equation is applied to spatially inhomogeneous states
near steady uniform shear flow. The shear rate of the reference steady state
can be large so the states considered include those very far from equilibrium.
The single particle distribution function is calculated exactly to first order
in the deviations of the hydrodynamic field gradients from their values in the
reference state. The corresponding non-linear hydrodynamic equaitons are
obtained and the set of transport coefficients are identified as explicit
functions of the shear rate. The spectrum of the linear hydrodynamic equation
is studied in detail and qualitative differences from the spectrum for
equilibrium fluctuations are discussed. Conditions for instabilities at long
wavelengths are identified and disccused.
