Turbulence unbound: unravelling anomalies, unpredictability, and flow breakdown – TURBO

Turbulence remains a major challenge for physicists, engineers, and mathematicians. The main issue is to develop a unified description of the limit of very high Reynolds numbers, which characterizes most practical situations. In this regime, turbulent flows acquire a sufficiently irregular structure to dissipate kinetic energy despite a viscosity approaching zero. This dissipative anomaly provides a consistent framework for selecting dynamic solutions. However, recent mathematical constructions of dissipative singular Euler solutions raise problems of non-uniqueness, questioning their relevance for describing turbulence and urging a re-examination of the notion of physical admissibility.
This project aims to resolve this dilemma. The first objective is to demonstrate that energy criteria are insufficient to select physically relevant turbulent solutions. The goal is to identify other invariants and anomalies, such as circulation and intermittency, and to demonstrate their universal nature, thereby providing new physical constraints. The second development is based on the recent concept of Eulerian spontaneous stochasticity: singular solutions to nonlinear equations exhibit explosive sensitivity to small perturbations. The project aims to prove that this phenomenon is generic and that non-uniqueness is an inherent characteristic of turbulence, challenging the very notion of a velocity field. Building on these ideas, we aim to reconcile the issue of non-uniqueness with the construction of solutions within a probabilistic framework, by developing a new (inviscid) formulation based on a generalized least-action principle. This work opens the way to a unified description of infinite-Reynolds-number turbulence and the development of innovative modeling tools.