欧拉方程在 $\mathbb{R}^3$ 上奇性的稳定性框架
Stability Framework for the Singularity of the Euler Equations on $\mathbb{R}^3$
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中文总结 AI 辅助
本文为 $\mathbb{R}^3$ 上欧拉方程的自相似奇异轮廓建立非线性稳定性证明框架，将分析归结为有限显式估计与常数验证，并支持重构为有限时间爆破的可容许解。
中文摘要 AI 辅助
在一项配套的数值研究中，我们在无界域 $\mathbb{R}^3$ 上发现了欧拉方程的一个高精度奇异轮廓。本文通过详细建立一个用于证明近似自相似轮廓（非线性）稳定性的框架来补充该研究，将分析归结为一大类但有限的显式估计和可计算常数的集合。在论证中出现的估计和常数得到严格验证的条件下，该框架完成了稳定性证明，并且至关重要的是，允许将稳定的重缩放轮廓重构为原始变量中的可容许解，该解在有限物理时间内变为奇异。随着稳定性和重构机制的建立，剩余的工作主要是定量的：验证显式常数，并在必要时改进选定的解析估计以完成稳定性证明。
英文摘要
In a recent numerical study, we found a high-precision singular profile for the Euler equations on the unbounded domain $\mathbb{R}^3$. The present manuscript complements that study by establishing in detail a preliminary framework for proving (nonlinear) stability of the approximate self-similar profile, reducing the analysis to a large but finite collection of explicit estimates and computable constants. Conditional on rigorous certification of the estimates and constants appearing in the argument, and on the candidate profile satisfying the required nonlinear stability conditions, the framework closes the stability proof and, crucially, allows the resulting stable rescaled profile to be reconstructed as an admissible solution in the original variables that becomes singular in finite physical time. With the overall stability and reconstruction mechanisms formulated, the remaining work within this approach is largely quantitative: determining whether the explicit constants and margins can be rigorously certified with sufficient positive margin and, where necessary, sharpening selected analytic estimates.
发表机构
- University of California Berkeley（加州大学伯克利分校）
- California Institute of Technology（加州理工学院）
机构由 AI 辅助整理，请以论文原文为准。
