Regularity and Approximation of Solutions to Conservation Laws

双曲守恒定律为连续介质物理提供了基本的数学模型，并被科学家和工程师广泛应用于交通流和火焰传播前沿的研究中。人们普遍期望这些方程应该是确定性的：知道一个初始配置，就应该能够唯一地预测未来的演变。然而，最近的数学进展指出，事实并非总是如此。该项目的一个主要目标是更好地理解在哪种情况下可以保证解的唯一性，而不是在一个或多个空间维度中出现多个解的例子。基于这些理论进展，研究人员将为广泛的计算方案提供新的误差界限，这些方案在应用中用作预测工具。一个进一步的研究方向将是准确地描述解是如何失去规律性的。换句话说：当一个新的冲击波，如压力的突然变化，形成的第一个瞬间会发生什么。该项目将为研究生和博士后提供研究培训机会。该项目将解决当前双曲守恒定律理论前沿的一些基本问题。新的唯一性或非唯一性结果将在更广泛的弱解类别中寻求，可能具有无界变分。对于具有严格凸熵的一维双曲守恒律，研究者旨在建立通用误差估计，该估计适用于所有与守恒方程和熵条件相容的近似格式。此外，对于各种类型的非线性波动方程，将在出现新奇点的点的邻域内提供一般解的局部渐近描述。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Hyperbolic conservation laws provide basic mathematical models for continuum physics, and are widely used by scientists and engineers, for instance, in the study of traffic flows and flame propagation fronts. There is a general expectation that these equations should be deterministic: knowing an initial configuration one should be able to uniquely predict the future evolution. However, recent mathematical advances point to the fact that this is not always true. A major goal of this project is to better understand in which situations the uniqueness of solutions can be guaranteed, compared with examples where multiple solutions occur, in one or more space dimensions. Based on these theoretical advances, the investigator will then provide new error bounds for a wide class of computational schemes, which are used in applications as predictive tools. A further research direction will be the accurate description of how solutions can lose regularity. In other words: what happens at the first instant of time when a new shock wave, such as a sudden alteration in pressure, is formed. The project will provide research training opportunities for graduate students and postdoctoral associates. The project will address some fundamental issues at the frontier of the current theory of hyperbolic conservation laws. New uniqueness or non-uniqueness results will be sought, in a wider class of weak solutions, possibly with unbounded variation. For one-dimensional hyperbolic conservation laws endowed with a strictly convex entropy, the investigator aims at establishing universal error estimates, valid for all approximation schemes which are compatible with the conservation equations and the entropy conditions. In addition, for various classes of nonlinear wave equations, a local asymptotic description of generic solutions will be provided, in a neighborhood of a point where a new singularity emerges.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.