二维不可压缩欧拉方程柯西问题所报道的多重解的数值研究
Numerical Study of Reported Multiple Solutions to a Cauchy Problem for the 2d Incompressible Euler Equations
AI总结：
本文通过精心设计的嵌套网格数值方法，研究二维不可压缩欧拉方程柯西问题中报道的非唯一弱解，发现仅一组参数对可能支持该论断。
AI中文摘要：
Bressan和Shen（2021）与Bressan和Murray（2020）合作提出了一个在$\mathbb{R}^2$上的柯西问题的简单例子，该问题由两个关键参数刻画，旨在产生不可压缩欧拉方程的弱解非唯一性。他们提出，对示例初始数据的两个特定近似应用极限过程会产生两个不同的解。在此，我们通过精心设计的数值程序提供证据，以支持这一引人注目的论断是否成立以及何时成立。本研究中遇到的一些数值挑战包括：（i）所提出的初始数据在原点邻域内无界，（ii）所提出的问题定义在整个$\mathbb{R}^2$上，（iii）必须捕获真实解中存在的几个基本对称性，（iv）所提出的示例依赖于两个可能关键相关的参数。我们证明，一个极大的计算域在数学上是必要的，以捕获精确自由空间问题的核心对称性。因此，我们实现了一种嵌套网格策略，该策略专门针对当前问题具有特别的适用性和有效性。我们的策略还使我们能够在解精度最关键的区域内实现极端的网格细化。在我们的嵌套网格框架内，我们设计了一种高阶、稳健且快速的底层离散格式。我们考虑了广泛的允许参数对，但在所有参数对中，我们只发现一个可能满足所提出的论断。
英文摘要：
Bressan and Shen (2021) jointly with Bressan and Murray (2020) proposed a simple example of a Cauchy problem on $\mathbb{R}^2$, characterized by two key parameters, intended to yield nonunique weak solutions of the incompressible Euler equations. They suggest that a limiting process applied to two specific approximations of their example initial data yields two distinct solutions. Here we provide evidence coming from a carefully designed numerical procedure to support if, or when, this intriguing claim is valid. Some of the numerical challenges encountered in the present study are: (i) the proposed initial data is unbounded in a neighborhood of the origin, (ii) the proposed problem is posed on all of $\mathbb{R}^2$, (iii) several essential symmetries present in the true solution must be captured, (iv) the proposed example depends on two parameters which may be critically linked. We demonstrate that an extremely large computational domain is mathematically necessary to capture the core symmetries of the exact free-space problem. Accordingly we implement a nested-grid strategy specifically designed for its particular suitability and efficacy for the problem at hand. Our strategy also allows us to achieve extreme grid refinement in the region where solution accuracy is most critical. Within our nested-grid framework, we design an underlying discretization scheme that is high-order, robust, and fast. We consider a broad collection of allowable parameter pairs, but among all of them we find only one that may satisfy the suggested claim.
