The maximum principle for the equation of the continuity of a compressible medium

From the text (translated from the Russian): "Two-sided estimates for the density rho (x,t) in terms of the initial state rho _0(x) and the flow v(x,t), having the character of a maximum principle, are useful when solving initial-boundary value problems of the dynamics of a compressible continuum. For a bounded domain Omega subset {bf R}^n, ngeq 2, we consider both classical and generalized solutions of the problem (1) partial rho /partial t+{rm div}(rho v)=0, xin Omega , t>0, (2) rho vert _{t=0}=rho _0(x), (v,nu )vert _{partial Omega }=0, for a given flow v(x,t) with a condition for the nonpenetration of partial Omega , where nu is the unit normal to partial Omega . We present conditions on the flow v(x,t), sufficient for a maximum principle to hold, whose accuracy we illustrate with examples of flows for which the maximum principle makes no sense, namely: either the density rho (x,t) vanishes (i.e., an expanding bubble forms in the continuum) or a generalized solution of problem (1), (2) does not even exist."