Pathological solutions of fluid equations

University Of Illinois At Chicago, Chicago IL

Investigators

Abstract

This project studies some fundamental open questions concerning the equations of fluid motion. The equations, widely used by physicists and engineers for real-life applications, were introduced almost two centuries ago. However, some essential questions, such as the existence and uniqueness of classical solutions, are still not answered. Therefore, a notion of weak solutions, whose existence is known, has been extensively used by mathematicians. The project will demonstrate a limitation of this notion by constructing weak solutions with various unphysical properties. On the other hand, the project proposes to prove the existence of physical solutions that are expected to describe turbulent flows. This project will also provide opportunities for the involvement of graduate students in the project. In the past couple of decades, mathematical fluid dynamics has been highlighted by numerous constructions of solutions to fluid equations that exhibit pathological or wild behavior, such as loss of the energy balance, non-uniqueness, singularity formation, and dissipation anomaly. Interesting from the mathematical point of view, these solutions provide counterexamples to various well-posedness results in supercritical spaces. Moreover, these constructive approaches are becoming more and more relevant from the physical point of view as well. Indeed, a fundamental physical property of turbulent flows is the existence of the energy cascade. Conjectured by Kolmogorov, the energy cascade has been observed both experimentally and numerically but had been difficult to produce analytically. The purpose of this project is to use state-of-the-art methods, such as convex integration and new developments in efficient mixing, to construct not only mathematically pathological, but also physically realistic solutions of fluid equations. 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.