The investigator addresses the qualitative properties of solutions to the Navier-Stokes, Euler, quasi-geostrophic, and other important equations in fluid dynamics. He studies nonlinear problems from meteorology, oceanography, and fluid mechanics, using state of the art mathematical methods from analysis and the theory of nonlinear partial differential equations. He continues his investigation of regularity theory in 3D fluid mechanics. A particular emphasis is the consequences of the non-conservation of circulation for the Navier-Stokes equation compared to the Euler equation. Fractional diffusion operators are crucial to model turbulent motion of passive or active scalars, or hysteresis phenomena (with memory). He develops a systematic study of this family of equations.
The investigator focuses on mathematical models involved in fluid mechanics. The Navier-Stokes and Euler equations are fundamental systems to study the dynamics of fluids. The understanding of behavior of the solutions is of fundamental importance in engineering, meteorology, medicine, and oceanography. When tracking the behavior of atmospheric phenomena, crucial for the study of climate change, those equations prove too complex for analytical treatment. The quasi-geostrophic equation is a simplified equation, derived from the fundamental equations, that still captures important dynamics at large scales of the flow. The investigator undertakes a systematic study of this important equation, which is commonly used in oceanography.