ERC Starting Grant - GA 101219635 - MIND
Mathematical Insights into Dynamics of Incompressible Turbulence
Principal Investigator: Elia Bruè
Abstract: The main goal of the MIND project is to advance the study of mathematical models describing turbulent evolutions. Beyond “classical chaos”, turbulence displays genuinely anomalous behavior that cannot be captured within classical well-posedness frameworks for PDEs. In fact, the physically meaningful regimes often live in very singular/flexible classes, where solutions may exist only in a weak sense and can even fail to be unique.
From the point of view of applications, this is a critical matter: turbulence sits at the core of engineering and technology (aerodynamics, energy, weather and ocean modelling, mixing and combustion). As Feynman famously put it, it remains “the most important unsolved problem of classical physics.” Any progress in the mathematical description has the potential to make modelling more reliable, inform principled simplifications, and ultimately improve the accuracy and robustness of computational predictions.
The flagship mathematical model in this area is the Navier–Stokes equation. A central open problem is the Navier–Stokes existence and smoothness Millennium Prize Problem: one of the seven Millennium Problems announced by the Clay Mathematics Institute in 2000 (with a $1,000,000 prize), asking whether 3D incompressible Navier–Stokes solutions remain smooth for all time or can develop singularities.
Mathematically, the study of turbulence models is a deep challenge that forces us to study linear and nonlinear PDE outside the comfort zone of classical well-posedness. The long-term aim is to connect modern PDE tools (DiPerna–Lions theory, dynamical-systems/Pesin ideas, and convex-integration technology) with the major phenomenological pictures of Onsager and Kolmogorov/Obukhov–Corrsin - and recent breakthroughs suggest that this bridge is finally becoming buildable.