Analysis Seminar

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Lagrangian chaos and scalar advection in stochastic fluid mechanics

Speaker: Jacob Bedrossian, Univ. of Maryland

Location: Warren Weaver Hall 1302

Date: Thursday, November 1, 2018, 11 a.m.

Synopsis:

We study the Lagrangian flow associated to velocity fields arising from various models of fluid mechanics 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. Our main results are for the 2D Navier-Stokes equations and the hyper-viscous regularized 3D Navier-Stokes equations (at arbitrary Reynolds number and hyper-viscosity parameters). As an application, we study statistically stationary solutions to the passive scalar advection-diffusion equation driven by these velocities and subjected to random sources. The chaotic Lagrangian dynamics are used to prove a version of anomalous dissipation in the limit of vanishing diffusivity, which in turn, implies that the scalar satisfies Yaglom's 1949 law of passive scalar turbulence -- the analogue of the Kolmogorov 4/5 law. The work combines ideas from random dynamical systems (the Multiplicative Ergodic Theorem and a version of Furstenberg's Criterion) with elementary approximate control arguments and hypoellipticity via Malliavin calculus. Joint work with Alex Blumenthal and Sam Punshon-Smith.