Chaos in stochastic 2d Galerkin-Navier-Stokes

Abstract

We prove that all Galerkin truncations of the 2d stochastic Navier-Stokes equations in vorticity form on any rectangular torus subjected to hypoelliptic, additive stochastic forcing are chaotic at sufficiently small viscosity, provided the frequency truncation satisfies N ≥ 392. By "chaotic" we mean having a strictly positive Lyapunov exponent, i.e. almost-sure asymptotic exponential growth of the derivative with respect to generic initial conditions. A sufficient condition for such results was derived in previous joint work with Alex Blumenthal which reduces the question to the non-degeneracy of a matrix Lie algebra implying Hörmander's condition for the Markov process lifted to the sphere bundle (projective hypoellipticity). The purpose of this work is to reformulate this condition to be more amenable for Galerkin truncations of PDEs and then to verify this condition using a) a reduction to genericity properties of a diagonal sub-algebra inspired by the root space decomposition of semi-simple Lie algebras and b) computational algebraic geometry executed by Maple 1 in exact rational arithmetic. Note that even though we use a computer assisted proof, the result is valid for all aspect ratios and all sufficiently high dimensional truncations; in fact, certain steps simplify in the formal infinite dimensional limit.

Lower bounds on the Lyapunov exponents of stochastic differential equations

“…In this article we will review existing work and our recent contributions [16,21] in proving that a given system of interest modeled by a stochastic differential equation is chaotic as in high sensitivity to initial conditions for trajectories initiated at Lebesgue-typical points in phase space. The specific systems we apply our methods to are the Lorenz-96 system [67] and Galerkin truncations of the 2d Navier-Stokes equations in a rectangular, periodic box (of any aspect ratio), provided they are subjected to sufficiently strong stochastic forcing 1 (equivalently, sufficiently weak damping) and are sufficiently high dimensional.…”

“…Section 2 concerns formulae of Lyapunov exponents through the stationary statistics of tangent directions and contains both classical results and our recent results from [16] which connects Lyapunov exponents to a certain Fisher information-type quantity. 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].…”

“…The term −ǫA provides weak linear damping, where A is assumed to be a symmetric, positive-definite n × n matrix. Stochastically forced versions of the Lorenz 96 model (L96) [67], Galerkin truncations of 2d and 3d Navier-Stokes on a torus (of arbitrary aspect ratio) [21,37,78] and truncations of commonly used shell models for turbulence [35,44,68,84] can be cast in this form. The 2D stochastic Galerkin-Navier-Stokes equations will be described in more detail in Section 4.3 below.…”

“…where u is the divergence free velocity field satisfying the Biot-Savart law u = ∇ ⊥ (−∆) −1 w and Ẇt is a white-in time, colored-in-space Gaussian forcing which we will take to be diagonalizable with respect to the Fourier basis with Fourier transform supported on a small number of modes. In the work [21] by the first and last authors of this note, we consider a Galerkin truncation of the 2d stochastic Navier-Stokes equations at an arbitrary frequency N ≥ 1 in Fourier space by projecting onto the Fourier modes in the truncated lattice…”

“…We will regard the configuration space C Z 2 0,N as a complex manifold with complexified tangent space spanned by the complex basis vectors {∂ w k : k ∈ Z 2 0,N } (Wirtinger derivatives) satisfying ∂ w −k = ∂w k . See [52] for the notion of complexified tangent space and [21] for discussion on how to use this complex framework for checking Hörmander's condition. In this basis, we can formulate the SDE (4.6) in the canonical form…”