This paper addresses the construction and the stability of self-similar solutions to the isentropic compressible Euler equations. These solutions model a gas that implodes isotropically, ending in a singularity formation in finite time. The existence of smooth solutions that vanish at infinity and do not have vacuum regions was recently proved and, in this paper, we provide the first construction of such smooth profiles, the first characterization of their spectrum of radial perturbations as well as some endpoints of unstable directions. Numerical simulations of the Euler equations provide evidence that one of these endpoints is a shock formation that happens before the singularity at the origin, showing that the implosion process is unstable.

本文讨论了等熵可压缩欧拉方程自相似解的构造和稳定性。这些解决方案模拟了一种气体，该气体各向同性内爆，在有限时间内以奇点形成结束。最近证明了在无穷远处消失并且没有真空区域的光滑解的存在，在本文中，我们提供了这种光滑轮廓的第一个构造，它们的径向扰动谱的第一个特征以及不稳定的一些端点方向。欧拉方程的数值模拟提供的证据表明，这些端点之一是发生在原点奇点之前的激波形成，表明内爆过程是不稳定的。