标题： 超音速引力坍缩对于非等熵气体恒星 显示英文标题

标题： Supersonic Gravitational Collapse for Non-Isentropic Gaseous Stars

摘要： 我们展示了非等熵欧拉-泊松系统中一类新的初始光滑球对称自相似解的存在性。 这些解在密度在有限时间内爆炸的同时流体速度保持超音速的意义上表现出超音速引力坍缩。 特别是，它们占据了远离最近构造的等熵自相似坍缩的部分相空间。 我们证明的核心在于存在一个两参数缩放不变性，并将问题简化为一个非自治常微分方程组。 我们利用流动的光滑性作为选择原则，限制了缩放指数的选择。 我们分析的一个重要结果是，对于我们构造的所有解，多普勒指数$\gamma$严格大于$\frac{4}{3}$，这与等熵情况下的已知结果形成鲜明对比。 显示英文摘要

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.