Chaos in stochastic 2d GalerkinNavierStokes
Abstract
We prove that all Galerkin truncations of the 2d stochastic NavierStokes 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. almostsure 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 nondegeneracy 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 subalgebra inspired by the root space decomposition of semisimple 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.
 Publication:

arXiv eprints
 Pub Date:
 June 2021
 arXiv:
 arXiv:2106.13748
 Bibcode:
 2021arXiv210613748B
 Keywords:

 Mathematics  Probability;
 Mathematics  Analysis of PDEs;
 Mathematics  Dynamical Systems;
 Physics  Fluid Dynamics
 EPrint:
 40 pages, 3 figures