Lagrangian chaos and scalar advection in stochastic fluid mechanics

Abstract

We study the Lagrangian flow associated to velocity fields arising from various models of stochastic fluid mechanics. We prove that in many circumstances, these flows are chaotic, that is, the top Lyapunov exponent is strictly positive (almost surely, all particle trajectories are simultaneously exponentially sensitive with respect to initial conditions). Our main results are for the Navier-Stokes equations on T 2 and the hyper-viscous regularized Navier-Stokes equations on T 3 (at arbitrary fixed Reynolds number and hyper-viscosity parameters), subject to white-in-time, H s -in-space stochastic forcing which is nondegenerate at high frequencies. Using these results, we further make a mathematically rigorous study of "passive scalar turbulence". To this end, we study statistically stationary solutions to the passive scalar advection-diffusion advected by one of these velocity fields and subjected to a white-in-time random source. We show that the chaotic behavior of Lagrangian dynamics implies a type of anomalous dissipation in the limit of vanishing diffusivity, which in turn, implies Yaglom's law of scalar turbulence -the universal scaling law analogous to the Kolmogorov 4/5 law. Key features of our study are the use of tools from ergodic theory and random dynamical systems in infinite dimensions, namely the Multiplicative Ergodic Theorem and a version of Furstenberg's Criterion, combined with hypoellipticity via Malliavin calculus and approximate control arguments.

The Batchelor Spectrum of Passive Scalar Turbulence in Stochastic Fluid Mechanics at Fixed Reynolds Number

“…Previously Lagrangian chaos has been proved for stationary, white-in-time velocity fields in [13,15]. In [16], we proved that the Lagrangian flow associated to solutions u t of the stochastically forced 2D Navier-Stokes and 3D hyperviscous Navier-Stokes is chaotic in the sense that there is a deterministic constant 1 > 0 such that (1.2) holds with lim-sup replaced with lim for all x and all initial fluid configurations, almost surely (see [16] for rigorous statements).…”

Sufficient conditions for local scaling laws for stationary martingale solutions to the 3D Navier-Stokes equations

“…Second, the conditions that we have assumed our solutions of (NSE) to satisfy seem very hard to verify. However, see [2] for a proof of a weak anomalous dissipation condition for the problem of passive scalar advection. At any rate, provided that one can obtain relations similar to the ones we derive in the next section, we believe that similar steps can be taken to derive local cascade laws for other systems that are expected to exhibit this sort of behavior, for instance one could attempt to adapt the derivations of the scaling laws in [4] or [2] to bounded domains in the same way that our paper adapts the results in [3].…”

Large Deviations and Entropy Production in Viscous Fluid Flows

“…Applying Theorem 5.1, one can prove that M is the unique stationary measure for (1.12). Let us note that the uniqueness of a stationary distribution was proved in [BBP18] for the coupled system (1.1), (0.6) with a coloured white noise η; however, their approach is not applicable in our situation since it is based on the strong Feller property and requires the noise to be rough in the space variables.…”

“…This type of results is not sufficient to get the convergence of the law of y to a limiting measure or to study the large deviations for empirical measures. We also mention the recent article [BBP18], which studies another aspect of chaotic behaviour of fluids-the strict positivity of the top Lyapunov exponent for the dynamics of the Lagrangian particle. The hypotheses imposed in [BBP18] are somewhat different from ours and require the noise to be sufficiently irregular in the space variables.…”