  In this paper, we study the global existence and low Mach number limit of
strong solutions to the 2-D full compressible Navier-Stokes equations around
the plane Couette flow in a horizontally periodic layer with non-slip and
isothermal boundary conditions. It is shown that the plane Couette flow is
asymptotically stable for sufficiently small initial perturbations, provided
that the Reynolds number, Mach number and temperature difference between the
top and the lower walls are small. For the case that both the top and the lower
walls maintain the same temperature, we further prove that such global strong
solutions converge to a steady solution of the incompressible Navier-Stokes
equations as the Mach number goes to zero.
