We consider the 2D isentropic compressible Euler equations, with pressure law p(ρ) = (1γ)ρ^γ, with γ > 1. We provide an elementary constructive proof of shock formation from smooth initial data of finite energy, with no vacuum regions, and with nontrivial vorticity. We prove that for initial data which has minimum slope −1ε, for ε > 0 taken sufficiently small relative to the amplitude, there exist smooth solutions to the Euler equations which form a shock in time . The blowup time and location can be explicitly computed and solutions at the blowup time are of cusp-type, with Hölder C¹³ regularity.

中文翻译：

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二维等熵可压缩欧拉的激波形成

我们考虑二维等熵可压缩欧拉方程，具有压力定律p ( ρ ) = (1 γ ) ρ ^γ，其中γ  > 1。我们根据有限能量的平滑初始数据提供了冲击形成的基本构造证明，没有真空区域，并且具有非平凡的涡度。我们证明，对于具有最小斜率 -1 ε的初始数据，对于ε  > 0 相对于幅度取足够小的值，欧拉方程存在平滑解，其在时间上形成冲击。可以明确计算爆破时间和位置，并且爆破时间的解是尖点型的，使用 Hölder C ¹³规律性。