Supersonic Gravitational Collapse for Nonisentropic Gaseous Stars

We show the existence of a new class of initially smooth spherically symmetric self-similar solutions to the non-isentropic Euler--Poisson system. These solutions exhibit supersonic gravitational implosion in the sense that the density blows-up in finite time while the fluid velocity remains supersonic. In particular, they occupy a portion of the phase space that is far from the recently constructed isentropic self-similar implosion. At the heart of our proof is the presence of a two-parameter scaling invariance and the reduction of the problem to a non-autonomous system of ordinary differential equations. We use the requirement of smoothness of the flow as a selection principle that constrains the choice of scaling indices. An important consequence of our analysis is that for all the solutions we construct, the polytropic index $\ga$ is strictly bigger than $\frac{4}{3}$, which is in sharp contrast to the known results in the isentropic case.