可压缩欧拉方程的结构稳定奇性与Lipschitz稳定最优输运度量
Structurally stable singularities and Lipschitz stable optimal transport metrics for the compressible Euler equations
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中文总结 AI 辅助
本文研究一维可压缩欧拉方程首次奇性：证明一般初值下解在有限点外二次可微且奇性为Hölder指数1/3的尖点，并构造Finsler度量使解Lipschitz连续依赖初值，从而验证该奇性的结构稳定性。
中文摘要 AI 辅助
众所周知，可压缩欧拉方程的解可以在有限时间内发展出奇性。本文对一维可压缩欧拉方程在一般光滑初值下，直至首次奇性发生时刻的解的行为进行了详细分析。我们的主要结果包含三个部分。首先，对于$C^3$初值的一个开稠密集合，利用Thom横截性定理，我们证明当首次奇性发生时，除至多有限个点外，欧拉方程的解是二次连续可微的。其次，对于第一部分给出的$C^3$函数开稠密集合中的任意初值，我们在首次奇性时刻每个奇点在$(x,t)$平面中的半邻域内，提供了解的精确渐近描述，并验证了解在每个奇点处具有Hölder指数$1/3$的尖点型奇性。前两个结果的证明基于用半线性系统表示的解。第三，对于具有小BV范数的光滑初值，我们构造了两个Finsler型最优输运度量，然后在这些度量下证明解在首次奇性时刻之前Lipschitz连续依赖于初值，且Lipschitz常数一致有界。特别地，$C^{1/3}$一般奇性在此意义下是稳定的。另一方面，由于我们的前两个结果对初值的开稠密集合成立，任何指数不同于$1/3$的Hölder连续尖点奇性在初值扰动下是不稳定的。
英文摘要
It is well known that solutions to the compressible Euler equations can develop singularities in finite time. In this paper, we carry out a detailed analysis on behaviors of solutions up to the time of the first singularity for the one-dimensional compressible Euler equations with general smooth initial data. Our main results consist of three parts. First, for an open dense set of $C^3$ initial data, we show that the solution of Euler equations is twice continuously differentiable except at most finitely many points when the first singularity happens, using Thom's Transversality Theorem. Second, for any initial data in the open dense set of $C^3$ functions given in the first result, we provide the precise asymptotic description of the solution in a semi-neighborhood in the $(x,t)$-plane of each singular point at the time of the first singularity, and verify that the solution has a cusp-type singularity with Hölder exponent $1/3$ at each singular point. The proofs of the first two results are based on the representation of the solution in terms of a semilinear system. Third, for smooth initial data with small BV norm, we construct two Finsler type optimal transport metrics, then under these metrics show that the solution depends Lipschitz continuously on the initial data up to the time of the first singularity, with uniformly bounded Lipschitz constants. In particular, the $C^{1/3}$ generic singularity is stable in this sense. On the other hand, since our first two results hold for an open and dense set of initial data, any Hölder continuous cusp singularity with exponent other than $1/3$ is unstable under initial perturbations.
发表机构
- University of Kansas（堪萨斯大学）
- Zhejiang University of Science and Technology（浙江科技大学）
机构由 AI 辅助整理，请以论文原文为准。
