Stability Theory for Systems of Hyperbolic Conservation Laws

该项目旨在开发数学工具来研究双曲守恒定律的稳定性理论，双曲守恒律对包括交通流和流体和气体动力学在内的各种物理系统进行建模。这些系统经常产生冲击或不连续，这对它们的数学处理构成了巨大的挑战。主要目的是研究这些模型的粘性近似的一致稳定性，特别是Navier-Stokes方程，以及在流体动力学中减轻粘性对激波的失稳影响的设计方法。该项目还将为本科生和研究生以及博士后研究人员提供培训和指导机会，以增强他们在建模、分析和交流方面的专业知识。这个项目将进一步发展双曲型守恒律的理论，通过推广带移位的加权压缩理论，得到弱/BV原理，BV解关于狂野初始扰动的稳定性，以及物理粘性模型的无粘性极限作为N-S方程。该项目将建立在加权收缩和移位理论的最新发展的基础上，以解决在等熵流动的情况下20年来的一个猜想。该项目还将考虑多维设置，其中解决方案的唯一性已知失败。虽然湍流预计会导致不稳定，但缺乏唯一性严重质疑模型本身的预测能力。这一病理给该领域带来了机遇，也带来了严峻的挑战。这项研究将开发一种理论来协调高雷诺数下不连续流动的不稳定性和可预测性。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。

This project aims to develop mathematical tools for studying the stability theory of hyperbolic conservation laws, which model a wide range of physical systems, including traffic flow and fluid and gas dynamics. These systems often develop shocks or discontinuities, which pose significant challenges for their mathematical treatment. The primary objective is to investigate the uniform stability of viscous approximations of these models, particularly the Navier-Stokes equation, and design methods to mitigate the destabilizing effect of viscosity on shocks in fluid dynamics. The project will also offer training and mentorship opportunities for undergraduate and graduate students and postdoctoral researchers, to enhance their expertise in modeling, analysis, and communication. This project will further develop the theory for hyperbolic conservation laws, by extending the theory of weighted contraction with shifts, to obtain weak/BV principles, stability of BV solutions with respect to wild initial perturbations, and inviscid limit of physical viscous models as the Navier-Stokes equation. The project will build on recent developments in the theory of weighted contraction with shifts to solve a twenty-year-old conjecture in the case of isentropic flows. The project will also consider multi-D settings where the uniqueness of solutions is known to fail. While instabilities are expected due to turbulence, the lack of uniqueness seriously questions the prediction abilities of the models themselves. This pathology brings both opportunities and formidable challenges to the field. This research will develop a theory to reconcile instability and predictability for discontinuous flows at high Reynolds numbers.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.