Development of Non-Perturbative Approaches to Partial Differential Equations Arising in Physical Applications

众所周知，更好地理解非线性过程是许多科学和技术领域取得进展的关键，包括气候预测和先进材料的开发。非线性科学的理论进步依赖于在复杂问题中推进数学理论和计算工具的前沿。计算工具虽然通常适用于非线性问题，但通常缺乏数学严谨性。数学严谨性在验证应用程序中使用的模型时非常重要。 另一方面，严格的数学分析方法通常局限于特殊的、较简单的设置。该项目旨在弥合数学和基于计算机的方法之间的差距，用于数学物理和流体动力学的一类重要应用。该项目侧重于广泛的发展和完善的严格的非微扰方法发现的主要研究人员，以回答数学物理和流体动力学中出现的重要开放问题。这些问题包括：（a）超临界偏微分方程，特别是Yang-Mills理论和波映射中正则解或爆破解的稳定性;（B）在非微扰区域中粘性指进的定常解在不同正则化下的稳定性及其线性和非线性稳定性的研究;和（c）研究大振幅振荡平行流体流的稳定性。

It is well-recognized that better understanding of nonlinear processes is key to progress in many areas of science and technology, including climate prediction and development of advanced materials. Theoretical progress in nonlinear sciences depends on advancing the frontiers of theoretical and computational tools of mathematics in complex problems. Computational tools, while generally suitable for nonlinear problems, usually lack mathematical rigor. Mathematical rigor is important in validating the models used in applications. Rigorous methods of mathematical analysis are, on the other hand, usually restricted to special, simpler settings. This project seeks to bridge the gap between mathematics and computer-based approaches for a class of important applications from mathematical physics and fluid dynamics.The project focuses on an extensive development and refinement of rigorous nonperturbative methods discovered by the principal investigators to answer important open questions that arise in mathematical physics and fluid dynamics. The problems include, in particular, the following: (a) stability of regular or blow-up solutions in super-critical partial differential equations, especially in Yang-Mills theory and wave maps; (b) investigation of steady solutions in viscous fingering in nonperturbative regimes with different regularizations, and of their linear and nonlinear stability; and (c) investigation of the stability of oscillating parallel fluid flows of large amplitudes.