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

Ohio State University, The, Columbus OH

Investigators

Abstract

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.