  We consider the Cauchy problem with smooth data for compressible Euler
equations in many dimensions and concentrate on two cases: solutions with
finite mass and energy and solutions corresponding to a compact perturbation of
a nontrivial stationary state. We prove the blowup results using the
characteristics of the propagation of the solution in space and find upper and
lower bounds for the density of a smooth solution in a given region of space in
terms of the initial data. To solve the problems, we introduce a special family
of integral functionals and study their temporal dynamics.
