The investigator analyzes several nonlinear nonlocal partial differential equations that arise in mathematical fluid mechanics. The goal is to develop new mathematical tools towards progress on various open problems, such as the global regularity versus finite time blow-up of classical solutions, the behaviour of solutions in singular limit regimes, and other qualitative properties. The first part of the project is devoted to studying the regularity of active scalar equations. The work is mostly geared towards the analysis of the inviscid and the supercritically-dissipative surface quasi-geostrophic equation, porous media equation, and the magneto-geostrophic equation. For all these models the a priori controlled quantities are too weak to guarantee the global existence of smooth solutions. The investigator and collaborators seek to identify analytic and geometric mechanisms that preclude the possible formation of singularities. The regularity of supercritical drift-diffusion equations is also considered. The second part of the project addresses the inviscid limit of the Navier-Stokes equations on a bounded domain. The goal is to establish new well-posedness results for the Prandtl boundary layer equations under weak matching conditions at the top of the boundary layer, and analyze the inviscid limit of the Navier-Stokes equations in these regimes.
The equations describing incompressible Newtonian fluids, such as the Euler and the Navier-Stokes equations, account for interactions between a very broad range of space and time scales, in a highly nonlinear fashion. Due to this intrinsic complexity, the accuracy needed to fully resolve the underlying phenomena via numerical computations is out of reach for the foreseeable future. This project seeks to further advance the rigorous analysis of these equations, which is vital to a finer understanding and validation of the models, and to an accurate interpretation of numerical simulations.
