权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

Singularities and Error Bounds for Hyperbolic Equations

双曲方程的奇点和误差界

基本信息

批准号：
2006884
负责人：
Alberto Bressan
金额：
$ 35.67万
依托单位：
Pennsylvania State Univ University Park
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-08-01 至 2023-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2006884&HistoricalAwards=false
关键词：
Singularities Error Bounds Hyperbolic Equations

项目摘要

The Euler equations, introduced in 1755, provide a basic mathematical description of the motion of fluids. While these equations are now extensively used in physics and engineering, some fundamental theoretical issues have remained unsolved. An outstanding open question is whether this model is deterministic. In other words, knowing the present configuration of a fluid, under which conditions can we uniquely predict its future behavior? Recent numerical experiments, performed by the principal investigator and collaborators, have identified certain initial states of the fluid which lead to multiple solutions. The present project will investigate the basic mechanism for which Euler's equations may fail to determine a unique solution. Specific examples will be studied, containing one or more spiraling vortices, to understand whether a similar loss of uniqueness occurs for compressible as well as incompressible fluid flow. The analysis will be carried out by a combination of theoretical and computational techniques. Rigorous estimates will be derived on the difference between a numerically computed approximation and the corresponding exact solution. The project will provide a training ground for various graduate students and young researchers.This project will investigate singularities of solutions to nonlinear wave equations. In particular, the analysis will focus on a class of initial value problems for the Euler equations modeling a two-dimensional, inviscid, compressible, or incompressible fluid flow. Based on recent numerical simulations conducted by the PI and collaborators, initial data having an algebraic singularity at the origin are expected to provide the simplest examples of Cauchy problems with multiple solutions, thus revealing a fundamental obstruction toward the well-posedness of the governing equations. This analysis will be carried out by a combination of theoretical and computational techniques. In a neighborhood of a spiraling vortex singularity, the solution will be studied by a suitable transformation of variables. On a domain where the solution is smooth, rigorous a posteriori error bounds for the numerical approximations will be derived. A related project will seek a posteriori error bounds for discrete numerical schemes, such as the Lax-Friedrichs and the Godunov scheme, in the computation of entropy-weak solutions to one-dimensional hyperbolic systems of conservation laws.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

1755年引入的欧拉方程提供了流体运动的基本数学描述。虽然这些方程现在在物理和工程中得到了广泛的应用，但一些基本的理论问题仍然没有得到解决。一个悬而未决的问题是，这个模型是否是确定性的。换句话说，知道一种流体目前的形态，我们在什么条件下才能唯一地预测它未来的行为？最近由主要研究人员和合作者进行的数值实验已经确定了流体的某些初始状态，这些状态导致了多个解。本项目将研究欧拉方程可能无法确定唯一解的基本机制。将研究包含一个或多个螺旋旋涡的具体例子，以了解可压缩和不可压缩流体流动是否会发生类似的独特性丧失。分析将通过理论和计算技术相结合的方式进行。对数值计算的近似解和相应的精确解之间的差异将得到严格的估计。这个项目将为不同的研究生和年轻的研究人员提供一个培训基地。这个项目将研究非线性波动方程解的奇性。特别是，分析将集中在欧拉方程的一类初值问题上，该方程模拟二维、无粘、可压缩或不可压缩的流体流动。根据PI和合作者最近进行的数值模拟，在原点具有代数奇异性的初始数据有望提供具有多个解的柯西问题的最简单例子，从而揭示控制方程适定性的根本障碍。这一分析将通过理论和计算技术的结合来进行。在旋涡奇点附近，将通过适当的变量变换来研究解。在解是光滑的区域上，将得到数值近似的严格的后验误差界。一个相关的项目将在计算一维双曲型守恒律系统的熵弱解时寻求离散数值格式的后验误差界，如Lax-Friedrichs和Godunov格式。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。