Lagrangian chaos and scalar advection in stochastic fluid mechanics

Abstract

We study the Lagrangian flow associated to velocity fields arising from various models of fluid mechanics subject to white-in-time, H s -in-space stochastic forcing in a periodic box. We prove that in many circumstances, these flows are chaotic, that is, the top Lyapunov exponent is strictly positive. 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 Reynolds number and hyper-viscosity parameters), subject to forcing which is non-degenerate at high frequencies. 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 law of scalar turbulence -the analogue of the Kolmogorov 4/5 law. Key features of our study are the use of tools from ergodic theory and random dynamical systems, namely the Multiplicative Ergodic Theorem and a version of Furstenberg's Criterion, combined with hypoellipticity via Malliavin calculus and approximate control arguments.

Lower bounds on the Lyapunov exponents of stochastic differential equations

“…In [17] we proved, under the condition that |q k | ≈ |k| −α for some α > 10, that ∃λ 1 > 0 deterministic and independent of initial x and initial velocity u such that the following limit holds almost-surely:…”

“…We discuss in Section 3 how to connect the Fisher information to regularity using ideas from hypoellipticity theory (also original work from [16]), and in Section 4 we discuss applications to a class of weakly-driven, weakly-dissipated SDE with bilinear nonlinear drift term (original work in [16] for Lorenz-96 and for Galerkin Navier-Stokes in [21]). In Section 5 we briefly discuss our earlier related work on Lagrangian chaos in the (infinite-dimensional) stochastic Navier-Stokes equations [17]. Finally, in Section 6 we discuss some open problems and potential directions for research.…”

“…In the volume preserving case however, criteria à la Furstenberg can be a very powerful tool. In our previous work [17], we used a suitable (partially) infinite-dimensional extension of Theorem 2.5 to show that the Lagrangian flow map (i.e. the trajectories of particles in a fluid) is chaotic when the fluid evolves by the stochastically forced 2D Navier-Stokes equations (called Lagrangian chaos in the fluid mechanics literature).…”

“…The main step is to deduce an analogue of Theorem 2.5 for the Lagrangian flow map, using that while the Lagrangian flow map depends on an infinite dimensional Markov process, the Jacobian D x ϕ t ω,u itself is finite dimensional. This is done in our work [17] by extending Furstenberg's criterion to handle general linear cocycles over infinite dimensional processes in the same way that D x ϕ t ω,u depends on the sample paths (u t ).…”

Enhanced dissipation for stochastic Navier-Stokes equations with transport noise

“…In the stochastic setting, L. Arnold et al [3] proved in the 1980s that suitable noises stabilize some finite dimensional linear ODE with a coefficient matrix of negative trace, see also [2]. In a series of recent papers [4,5,6], Bedrossian et al have shown the properties of exponential mixing and dissipation enhancement by flows which are solutions to stochastic Navier-Stokes equations, leading to a proof of Batchelor's conjecture on the spectrum of passive scalar turbulence [7]. We also mention [35] for results on stabilization and enhanced dissipation by transport noise, partly motivated by some ideas in [20,5].…”