Chaos in stochastic 2d Galerkin-Navier-Stokes

TL;DR

This paper proves that small viscosity Galerkin-truncated 2d stochastic Navier-Stokes equations on a torus exhibit chaos, characterized by positive Lyapunov exponents, using algebraic and computational methods to verify hypoellipticity conditions.

Contribution

It reformulates and verifies a hypoellipticity condition for chaos in Galerkin-truncated stochastic PDEs using algebraic geometry and computer algebra, extending previous theoretical frameworks.

Findings

Galilean truncations are chaotic at small viscosity with positive Lyapunov exponents.

Verification of hypoellipticity condition using algebraic geometry and Maple computations.

Results hold for all aspect ratios and high-dimensional truncations.

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 $N\geq 392$. 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 H\"ormander'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 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.