We consider the 3D isentropic compressible Euler equations with the ideal gas law. We provide a constructive proof of the formation of the first point shock from smooth initial datum of finite energy, with no vacuum regions, with nontrivial vorticity present at the shock, and under no symmetry assumptions. We prove that for an open set of Sobolev-class initial data that are a small L^∞ perturbation of a constant state, there exist smooth solutions to the Euler equations which form a generic stable shock in finite time. The blowup time and location can be explicitly computed, and solutions at the blowup time are smooth except for a single point, where they are of cusp-type with Hölder C^1/3 regularity. Our proof is based on the use of modulated self-similar variables that are used to enforce a number of constraints on the blowup profile, necessary to establish global existence and asymptotic stability in self-similar variables. © 2022 Wiley Periodicals LLC.

中文翻译：

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3D 可压缩欧拉点激波的形成

我们考虑具有理想气体定律的 3D 等熵可压缩欧拉方程。我们提供了从有限能量的平滑初始数据形成第一点激波的建设性证明，没有真空区域，激波处存在非平凡涡度，并且没有对称性假设。我们证明，对于一组开放的 Sobolev 级初始数据（恒定状态的小L ^∞扰动），欧拉方程存在光滑解，在有限时间内形成通用稳定激波。可以显式计算爆破时间和位置，并且爆破时的解除了单个点之外都是平滑的，其中它们是具有 Hölder 的尖点型C ^1/3规律性。我们的证明基于调制自相似变量的使用，这些变量用于对爆炸轮廓实施许多约束，这是建立自相似变量的全局存在性和渐近稳定性所必需的。© 2022 Wiley 期刊有限责任公司。