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，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。人们普遍期望这些方程应该是确定性的：知道一个初始构型，就应该能够唯一地预测未来的演化。然而，最近的数学进步表明，这并不总是正确的。这个项目的一个主要目标是，与在一个或多个空间维度中出现多个解决方案的例子相比，更好地了解在哪些情况下可以保证解决方案的唯一性。在这些理论进展的基础上，研究人员将为广泛的计算方案提供新的误差界，这些计算方案在应用中用作预测工具。进一步的研究方向将是对解决方案如何失去规律性的准确描述。换句话说：当一个新的冲击波形成时，第一个瞬间会发生什么，例如压力的突然变化。该项目将为研究生和博士后助理提供研究培训机会。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一类弱解中寻找，可能具有无界变化。对于具有严格凸熵的一维双曲守恒律，研究人员的目的是建立对所有与守恒方程和熵条件相容的逼近格式有效的普适误差估计。此外，对于不同类别的非线性波动方程，将在新奇点出现的附近提供一般解的局部渐近描述。该奖项反映了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。

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.