Chaos in Stochastic 2d Galerkin-Navier-Stokes

Area of Science:

• * Fluid dynamics
• * Stochastic partial differential equations
• * Chaos theory

Background:

• * The 2D stochastic Navier-Stokes equations model turbulent fluid flow with random influences.
• * Galerkin truncations approximate solutions by reducing the infinite-dimensional system to a finite one.
• * Previous work established a link between chaos and the non-degeneracy of a matrix Lie algebra.

Purpose of the Study:

• * To reformulate the non-degeneracy condition for easier application to Galerkin truncations.
• * To verify this condition for the 2D stochastic Navier-Stokes equations.
• * To establish chaos in truncated systems under specific conditions.

Main Methods:

• * Reformulation of a non-degeneracy condition for stochastic partial differential equations.
• * Application of Lie algebra properties, specifically root space decomposition.
• * Computational algebraic geometry using Maple for exact rational arithmetic verification.

Main Results:

• * All Galerkin truncations of the 2D stochastic Navier-Stokes equations exhibit chaotic behavior at small viscosity.
• * Chaos is defined as a strictly positive Lyapunov exponent, indicating exponential growth of solution derivatives.
• * The condition for chaos is satisfied provided the frequency truncation meets specific criteria.

Conclusions:

• * The study confirms chaotic dynamics in truncated 2D stochastic Navier-Stokes equations.
• * The findings are valid for all aspect ratios and sufficiently high-dimensional truncations.
• * Computer-assisted proof methods were employed, with potential simplifications in the infinite-dimensional limit.