  We consider the evolution of a quantity advected by a compressible flow and
subject to diffusion. When this quantity is scalar it can be, for instance, the
temperature of the flow or the concentration of some pollutants. Because of the
diffusion term, one expects the equations to have a regularizing effect.
However, in their Euler form, the equations describe the evolution of the
quantity multiplied by the density of the flow. The parabolic structure is thus
degenerate near vacuum (when the density vanishes). In this paper we show that
we can nevertheless derive uniform $L^{p}$ bounds that do not depend on the
density (in particular the bounds do not degenerate near vacuum). Furthermore
the result holds even when the density is only a measure. We investigate both
the scalar and the system case. In the former case, we obtain $L^{\infty}$
bounds. In the latter case the quantity being investigated could be the
velocity field in compressible Navier-Stokes type of equations, and we derive
uniform $L^p$ bounds for some $p$ depending on the ratio between the two
viscosity coefficients (the main additional difficulty in that case being to
deal with the second viscosity term involving the divergence of the velocity).
Such estimates are, to our knowledge, new and interesting since they are
uniform with respect to the density. The proof relies mostly on a method
introduced by De Giorgi to obtain regularity results for elliptic equations
with discontinuous diffusion coefficients.
