Supersonic Gravitational Collapse for Non-Isentropic Gaseous Stars
Christopher Alexander, Mahir Hadžić, Matthew Schrecker
TL;DR
This work constructs a new class of smooth, self-similar solutions to the non-isentropic Euler–Poisson system that exhibit supersonic gravitational implosion with finite-time central density blow-up. The authors exploit a two-parameter scaling and a smoothness selection principle to determine the self-similar exponents, establishing a gamma range ($\\gamma \in (19/12, 11/6)$) and fixing the scaling via $n(\gamma,\alpha)=4$. They prove local existence through convergent Taylor expansions around the sonic point and then extend these solutions globally using a bootstrap argument, showing the flow remains supersonic with a single sonic point at the origin and converges to a far-field state with $\omega(y) \to 2-\gamma$ and $\rho(y) \to 0$. The results illuminate how entropy transport enables supercritical implosions beyond the isentropic case, and provide precise mass-energy scaling relations consistent with the problem's invariances.
Abstract
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 $γ$ is strictly bigger than $\frac{4}{3}$, which is in sharp contrast to the known results in the isentropic case.