Workshop on New Ideas and Tools in Turbulence

Abstract: The purpose of this work is to perform a mathematically rigorous study of Lagrangian chaos and "passive scalar turbulence" in incompressible fluid mechanics. We study the Lagrangian flow map associated to 2D Navier-Stokes and hyper-viscous regularized 3D Navier-Stokes subject to white-in-time, Sobolev-in-space stochastic forcing in a periodic box. We prove that if the forcing satisfies suitable non-degeneracy conditions, then these flows are chaotic in the sense that the top Lyapunov exponent is strictly positive. For the passive scalar problem, we study statistically stationary solutions to the advection-diffusion equation driven by these velocities and subjected to random sources. The chaoticLagrangian dynamics are used to prove a variation of anomalous dissipation in the limit of vanishing diffusivity, which is then used to deduce that the scalar satisfies Yaglom's 1949 law over a suitable "inertial" range (the constant flux law analogous to the Kolmogorov 4/5 law). Thus, this work provides a complete mathematical proof of this scaling law. The work combines ideas from random dynamical systems (the Multiplicative Ergodic Theorem and an infinite dimensional variation of Furstenberg's Criterion) with elementary approximate control arguments and infinite-dimensional hypoellipticity via Malliavin calculus. Joint work with Alex Blumenthal and Sam Punshon-Smith.