A new type of stable shock formation in gas dynamics

Abstract

From an open set of initial data, we construct a family of classical solutions to the 1D nonisentropic compressible Euler equations which form $ C^{0,\nu} $ cusps as a first singularity, for any $ \nu \in [\frac{1}{2}, 1) $. For this range of $ \nu $, this is the first result demonstrating the stable formation of such $ C^{0,\nu} $ cusp-type singularities, also known as pre-shocks. The proof uses a new formulation of the differentiated Euler equations along the fast acoustic characteristic, and relies on a novel set of $ L^p $ energy estimates for all $ 1

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气体动力学中一种新型稳定激波形成

从一个开放的初始数据集，我们构造了一维非等熵可压缩欧拉方程的经典解族，这些方程形成$ C^{0,\nu} $顶点作为第一奇点，对于任何$ \nu \in [\frac{1}{2}, 1) $。对于这个$ \nu $范围，这是第一个证明这种$ C^{0,\nu} $尖型奇点(也称为预冲击)稳定形成的结果。该证明使用了沿快速声学特性的微分欧拉方程的新公式，并依赖于所有$ 1<p<\infty $的一组新的$ L^p $能量估计，这可能是独立的兴趣。

本文章由计算机程序翻译，如有差异，请以英文原文为准。