Invariant Measures of Gaussian Type for 2D Turbulence

Abstract

Gaussian measures μ β,ν are associated to some stochastic 2D models of turbulence. They are Gibbs measures constructed by means of an invariant quantity of the system depending on some parameter β (related to the 2D nature of the fluid) and the viscosity ν. We prove the existence and the uniqueness of the global flow for the stochastic viscous system; moreover the measure μ β,ν is invariant for this flow and is the unique invariant measure. Finally, we prove that the deterministic inviscid equation has a μ β,ν -stationary solution (for any ν > 0).

Langevin Dynamics, Large Deviations and Instantons for the Quasi-Geostrophic Model and Two-Dimensional Euler Equations

“…Even the expectation for the velocity field is not defined, and most of the realization do not lead to a physical velocity field. This is related to some of the mathematical results in [6]. These remarks give a negative answer to the first question.…”

“…This equation is a transversality property between the the vector field F and the gradient of the potential G. These two hypotheses, Liouville (5) and the conservation of the potential (6), are verified if the dynamical system is Hamiltonian:…”

“…We first define Langevin dynamics in subsection II A, as stochastic, ordinary or partial, differential equations, for which the deterministic part is composed of a vector field with a Liouville property (conservation of phase space volume, Eq. ( 5)) plus a potential force with potential G. The conservative part of the dynamics are assumed to be transverse to the gradient of the potential (6). The stochastic force is defined as the derivative of a Brownian process, with a correlation function identical to that of the kernel of the potential force.…”

“…What are the necessary and sufficient conditions for the formal computations performed in this work to be mathematically founded? Some of these questions are related to recent advances in the mathematics of stochastic partial differential equations [6,7,27,33,34,37,38]. Again, these aspects are further discussed in the conclusion.…”

Fokker-Planck equation for dissipative 2D Euler equations with cylindrical noise

“…The 2D Euler equations with additive noise, possibly including friction, their corresponding stationary solutions and invariant measures had already been considered before. However, the space regularity of noise is such that solutions are function-valued, not distributions and invariant measures are supported on spaces of functions: we refer for instance to [11,19,8,12,21,44,9,43,10], and also to other related results in [52,53,32]. Many of those models and results are inspired by the open problem of turbulence (iii); in connection with this question and the previous references we also mention [14,46,37].…”