Mathematical Fluid Mechanics

The first focus of our research is hydrodynamic turbulence, which Richard Feynman famously described as the most important unsolved problem in classical physics. We are interested in proving generic properties in incompressible turbulence. To this end, we develop certain multi-scale techniques specifically tailored to turbulence, combining convex integration, homogenization theory, and combinatorics. It allows us to formalize renormalization methods in hydrodynamic turbulence and to establish results consistent with the seminal Kolmogorov–Obukhov–Corrsin theory.

Our second line of research concerns regularity and singularity formation in fluid models. A typical problem in this area is the Millennium Prize Problem on the Navier–Stokes equations. On one hand, we investigate the degree of roughness exhibited by different fluid models; on the other, we aim to establish various permitted singular behavior. Ultimately, the goal is to deepen the understanding of notions such as solutions, uniqueness, and selection principles — questions that are closely intertwined with the turbulence problem.