Nonlocal equations in mathematical fluid dynamics

In this project, the PI aims to study partial differential equations (PDEs) arising from fluid dynamics. In a mathematical perspective, the fluid PDEs typically display nonlinear/nonlocal structure, which is the main source of difficulty in the analysis. Although such nonlinear/nonlocal structure is a ubiquitous feature in fluid PDEs, it reveals different phenomena in different situations, therefore different strategies must be taken accordingly in the analysis. For this reason, the PI proposes several research directions in which distinct behaviors of fluids will be investigated.Direction A) Steady state flow: A steady state flow is often given as a solution to an overdetermined elliptic PDE, which immediately implies that not every state can be a steady state. Steady fluids sometimes exhibit trivial structure but not always, and this feature strongly depends on the property of the nonlocal operator inherent to the equation. This sub-project aims to establish a necessary/sufficient condition for a steady fluid to enjoy non-trivial structure and discover its distinctive features. In particular, this research component will be focused on the two-dimensional incompressible Euler equation (2D Euler).Direction B) Fluids near a stable steady state: Tautologically, a smooth, steady fluid remains smooth for all time since it does not evolve non-trivially in time. A natural question is whether a small perturbation of such a smooth steady state can also remain smooth for all time. Especially, when the well-posedness of the governing equation is in question, the investigation of initial data near stable steady states can give insight into potential global-in-time solutions. In the second part of this proposal, the construction of global solutions to the generalized surface quasi-geostrophic equations (gSQG) is studied by means of KAM theory.Direction C) Long-time behavior of fluids: One of the fundamental questions in PDEs is whether general smooth initial data can develop a finite-time singularity or remain smooth for all time. Even if the solution remains smooth, it is often observed that its long-time behavior can be highly complicated. The main objective in this direction is to understand the long- time behavior of global solutions generated by the nonlinearity of the equations, especially instability of an equilibrium and a finite time singularity formation. The research project in this direction is centered around the two-dimensional incompressible Boussinesq system (2D Boussinesq), and its one-dimensional toy-model.The mathematical tools in this project vary from variational flavor techniques to microlocal analysis. The unifying goal of the whole project is to understand the role of nonlinearity/nonlocality in various situations of fluids, and develop/extend modern techniques to address related open problems.