Supersonic Gravitational Collapse for Non-Isentropic Gaseous Stars
arxiv(2023)
摘要
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 $\gamma$ is strictly bigger than $\frac{4}{3}$,
which is in sharp contrast to the known results in the isentropic case.
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