# Chaos in stochastic 2d Galerkin-Navier-Stokes

@inproceedings{Bedrossian2021ChaosIS, title={Chaos in stochastic 2d Galerkin-Navier-Stokes}, author={Jacob Bedrossian and Samuel Punshon-Smith}, year={2021} }

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…

## References

SHOWING 1-10 OF 60 REFERENCES

Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing

- Mathematics, Physics
- 2004

The stochastic 2D Navier-Stokes equations on the torus driven by degenerate noise are studied. We characterize the smallest closed invariant subspace for this model and show that the dynamics…

Lyapunov exponents and relative entropy for a stochastic flow of diffeomorphisms

- Mathematics
- 1989

SummaryThe Lyapunov exponents λ1≧λ2≧...≧λd for a stochastic flow of diffeomorphisms of a d-dimensional manifold M (with a strongly recurrent one-point motion) describe the almost-sure limiting…

Furstenberg's theorem for nonlinear stochastic systems

- Mathematics
- 1987

SummaryWe extend Furstenberg's theorem to the case of an i.i.d. random composition of incompressible diffeomorphisms of a compact manifold M. The original theorem applies to linear maps {Xi}i∈N on ℝm…

Quantitative spectral gaps and uniform lower bounds in the small noise limit for Markov semigroups generated by hypoelliptic stochastic differential equations

- Mathematics
- 2020

We study the convergence rate to equilibrium for a family of Markov semigroups $\{\mathcal{P}_t^{\epsilon}\}_{\epsilon > 0}$ generated by a class of hypoelliptic stochastic differential equations on…

Stochastic averaging and asymptotic behavior of the stochastic Duffing-van der Pol equation

- Mathematics
- 2004

Abstract Consider the stochastic Duffing–van der Pol equation x =−ω 2 x−Ax 3 −Bx 2 x +e 2 β x +eσx W t with A⩾0 and B>0. If β/2+σ2/8ω2>0 then for small enough e>0 the system (x, x ) is positive…

Algebraization of 2-D Ideal Fluid Hydrodynamical Systems and Their Finite-Mode Approximations

- Mathematics
- 1991

Starting from the description of ideal 2-D hydrodynamics in the framework of the Lie algebra of area-preserving diffeomorphisms sdiff two observations are made: 1st — this construction may be…

Ergodicity for the Navier-Stokes Equation with Degenerate Random Forcing: Finite-Dimensional Approximation

- Mathematics
- 2001

We study Galerkin truncations of the two-dimensional Navier-Stokes equation under degenerate, large-scale, stochastic forcing. We identify the minimal set of modes that has to be forced in order for…

Hypoelliptic second order differential equations

- Mathematics
- 1967

that is, if u must be a C ~ function in every open set where Pu is a C ~ function. Necessary and sufficient conditions for P to be hypoelliptic have been known for quite some time when the…

On Malliavinʼs proof of Hörmanderʼs theorem

- Mathematics
- 2011

The aim of this note is to provide a short and self-contained proof of Hormanderʼs theorem about the smoothness of transition probabilities for a diffusion under Hormanderʼs “brackets condition”.…

Finite-mode analogs of 2D ideal hydrodynamics: coadjoint orbits and local canonical structure

- Mathematics
- 1991

Abstract The algebraic finite-mode hydrodynamic-type systems that have O(N) integrals of motion for O(N × N) modes and are intrinsically connected with two-dimensional ideal fluid flows are studied.…