Topics in the Analysis of Nonlinear Partial Differential Equations

Some of the most useful models of natural phenomena use classical field theories and continuum mechanics. In these models, the objects aiming to describe reality are functions defined in regions of space-time and satisfying certain partial differential equations. The research carried out by the PI focuses on the equations arising in the context of fluid mechanics. The solutions of partial differential equations are usually described by an infinite number of parameters. This is a complication, but the reward is that the resulting models are in some sense canonical. For example, once we make a small number of very reasonable assumptions about fluid motion, we necessarily end up with the incompressible Navier-Stokes equation. There is essentially no ambiguity about what the model should be, and the Navier-Stokes equation is widely used in science and engineering, from airplane design, weather prediction, and climate modeling to computations of various water flows. To be able to perform computer simulations of the equations, it is necessary to reduce the continuum models to models described by finitely many parameters. However, this step is not canonical, there are many reasonable ways of doing it. When the continuum model is well-understood mathematically, its reductions to finite-dimensional models and their relations are relatively well understood. Of course, one should not overstate this - even in that case there are still many interesting open problems. However, our mathematical understanding of the Navier-Stokes equation is quite incomplete, and that makes the interpretations of the results from its finite-dimensional reductions harder. One way to think about the research in this project is that it aims to contribute to filling this gap in our knowledge and ultimately make our modeling more efficient. The project provides research training opportunities for graduate students.At a more technical level, this project focuses on the following topics: (i) Well-posedness, (non)-uniqueness, backward uniqueness, and critical situations for basic equations. In recent years, important advances have been made concerning well-posedness and non-uniqueness for the Navier-Stokes and related equations, but important basic questions still remain open. For example, can the solution operator be continuously extended from smooth solutions to spaces with topologies generated by natural physical quantities such as energy? Are regularity estimates in two-dimensional domains with boundary saturated by some solutions? (ii) Steady-state solutions of the three-dimensional Navier-Stokes equation and deformations of Serrin’s swirling vortex. The study of steady-state solutions, in addition to being of independent practical interest, provides insights for improving our understanding of time-dependent solutions. In this project, the focus is on deformations of certain known classes of solutions (due to Serrin) with symmetries to solutions with fewer symmetries. It is worth pointing out that both Serrin's solutions and the deformations envisaged here have connections to tornadoes; (iii) Liouville Theorems and related linear problems. Liouville theorems aim to describe global bounded solutions in all space-time and are closely related to open regularity questions about the equation. In this project, they are studied mostly in the steady-state setting. Their study also leads to interesting linear problems that will be addressed; (iv) One-dimensional models, including the study of possible avoidance of singularities by generic forcing for the quaternionic Burgers model and the global well-posedness and more detailed solution behavior for the De Gregorio model. There are many open problems even at the level of the one-dimensional models. These problems should provide good steppingstones towards improving our understanding and eventually applying the lessons learned from studying these models to higher dimensions.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.