  We investigate the long-time behavior of solutions to the isothermal Euler,
Korteweg or quantum Navier Stokes equations, as well as generalizations of
these equations where the convex pressure law is asymptotically linear near
vacuum. By writing the system with a suitable time-dependent scaling we prove
that the densities of global solutions display universal dispersion rate and
asymptotic profile. This result applies to weak solutions defined in an
appropriate way. In the exactly isothermal case, we establish the compactness
of bounded sets of such weak solutions, by introducing modified entropies
adapted to the new unknown functions.
