Singularities and Error Bounds for Hyperbolic Equations

1755年引入的欧拉方程提供了流体运动的基本数学描述。虽然这些方程现在广泛用于物理和工程，但一些基本的理论问题仍然没有解决。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，知道流体的当前构型，在什么条件下我们可以唯一地预测它的未来行为？最近的数值实验，由主要研究者和合作者进行，已经确定了某些初始状态的流体，导致多个解决方案。本项目将研究欧拉方程可能无法确定唯一解的基本机制。具体的例子将进行研究，包含一个或多个螺旋涡，以了解是否发生类似的损失的唯一性，以及不可压缩的流体流动。分析将结合理论和计算技术进行。严格的估计将得出数值计算的近似值和相应的精确解之间的差异。该项目将为各种研究生和年轻研究人员提供一个培训基地。该项目将研究非线性波动方程解的奇异性。特别是，分析将集中在一类初始值问题的欧拉方程建模的二维，无粘，可压缩，或不可压缩的流体流动。基于PI和合作者最近进行的数值模拟，在原点具有代数奇点的初始数据有望提供具有多个解的柯西问题的最简单示例，从而揭示了对控制方程适定性的根本障碍。这种分析将通过理论和计算技术相结合来进行。在螺旋涡奇点附近，将通过适当的变量变换来研究该解。在一个区域的解决方案是光滑的，严格的后验误差界的数值近似将被导出。一个相关的项目将寻求离散数值方案的后验误差界，如Lax-Friedrichs和Goddom方案，在计算一维双曲守恒律系统的熵弱解时。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

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.