Large Deviation Principle and Inviscid Shell Models

Abstract

A LDP is proved for the inviscid shell model of turbulence. As the viscosity coefficient ν converges to 0 and the noise intensity is multiplied by √ ν, we prove that some shell models of turbulence with a multiplicative stochastic perturbation driven by a H-valued Brownian motion satisfy a LDP in C([0, T ], V ) for the topology of uniform convergence on [0, T ], but where V is endowed with a topology weaker than the natural one. The initial condition has to belong to V and the proof is based on the weak convergence of a family of stochastic control equations. The rate function is described in terms of the solution to the inviscid equation.2000 Mathematics Subject Classification. Primary 60H15, 60F10; Secondary 76D06, 76M35. Key words and phrases. Shell models of turbulence, viscosity coefficient and inviscid models, stochastic PDEs, large deviations.

Invariant Gibbs measures of the energy for shell models of turbulence: the inviscid and viscous cases

“…Remark 3. 9 We point out that we have not proved that F C ∞ b is a core for the infinitesimal generator of the semigroup P t in L 1 µ ν , whereas we proved before (for any M ) that F C ∞ b is a core for the infinitesimal generator of the semigroup P M t in L 1 µ ν . The criterium of [30] used in the previous sections (with the quadratic term B M ) does not work for the operator K (with the "full" quadratic term B).…”

Invariant Measures of Gaussian Type for 2D Turbulence

“…Here, we present the details for the SABRA shell model (see [24]), but the same results hold for the GOY shell model (see [21,25]). In recent years, there has been an increasing interest in these fluid dynamical models, both for the deterministic and the stochastic case (see also [3,5,9,10]). From the analytic point of view as well as for numerical computations, they are easier to analyze than the Navier-Stokes or Euler equations.…”

Strong solutions to stochastic hydrodynamical systems with multiplicative noise of jump type

“…It is worth emphasizing that the presence of the stochastic term (noise) in these models often leads to qualitatively new types of behavior for the processes. Since the pioneering work of Bensoussan and Temam [4], there has been an extensive literature on stochastic Navier-Stokes equations with Wiener noise and related equations, we refer to [1], [2], [5], [6], [16], [19], [24], [42] amongst other.…”