Large time asymptotics of compressible Euler equations for a polytropic gas
with and without the porous media equation are constructed in which the
Barenblatt solution is embedded. Invariance analysis for these governing
equations are carried out using the classical and the direct methods. A new
second order nonlinear partial differential equation is derived and is shown
to reduce to an Euler–Painlev´e equation. A regular perturbation solution of
a reduced ordinary differential equation is determined. And an exact closed
form solution of a system of ordinary differential equations is derived using
the invariance analysis.