FLUTURA: Fluids, Turbulence, Advection

The Euler and the Navier-Stokes equations are basic mathematical models for the dynamics of incompressible fluids. Despite being among the first partial differential equations (PDEs) ever written, to a very large extent their mathematical theory remains a riddle. One of the main reasons is the intrinsic lack of regularity of solutions of problems in fluid dynamics. The theory of turbulence advances quantitative predictions for the "typical" properties of "chaotic" fluids, in particular with regard to universal behaviors at small spatial scales. These predictions are well supported by experimental observations, but very few is understood from the point of view of rigorous analysis. Understanding the correct framework validating and predicting the experimental evidence is an exciting and ambitious mathematical question.

In this project we address several challenging open problems in mathematical fluid dynamics. The common feature of all these problems is the appearance of irregular behaviors and the cascade of information to small spatial scales. The irregularity is the main source of difficulty in the mathematical analysis of these questions, since it precludes the use of most classical PDE methods, from energy estimates to the theory of characteristics.

We aim to derive a quantitative analysis and to provide detailed insight of the properties of solutions in physically relevant regimes. This project will significantly expand our theoretical understanding of several phenomena involving irregular flows, addressing in particular the following topics:

• Selection and nonselection by vanishing viscosity.
• Anomalous dissipation in the Obukhov-Corrsin theory of scalar turbulence.
• Conservation and dissipation of the kinetic energy and of the enstrophy in fluid flows.
• Mixing, regularity, and irregularity for fluid flows.
• Existence, uniqueness, and structure of solutions for nonlinear problems with diffuse/singular vorticity.

The unity of the project stems from the concepts and the methods in the background, all related to the theory of advection (both deterministic and stochastic) in a nonsmooth setting and relying in an essential way on techniques of geometric measure theory as a crucial tool in the analysis of the fine properties of irregular flows.