  We consider the interplay of thermal fluctuations and shear on the surface of
the domains in various systems coarsening under an imposed shear flow. These
include systems with nonconserved and conserved dynamics, and a conserved order
parameter advected by a fluid whose velocity field satisfies the Navier-Stokes
equation. In each case the equation of motion for the interface height reduces
to an anisotropic Burgers equation. The scaling exponents that describe the
growth and coarsening of the interface are calculated exactly in any dimension
in the case of conserved and nonconserved dynamics. For a fluid-advected
conserved order parameter we determine the exponents, but we are unable to
build a consistent perturbative expansion to support their validity.
