Some Rigorous Results on a Stochastic GOY Model

Barbato, David; Barsanti, Michele; Bessaih, Hakima; Flandoli, Franco

doi:10.1007/s10955-006-9203-y

Cited by 36 publications

(82 citation statements)

References 23 publications

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“…Thus, this result contains the corresponding existence and uniqueness theorems and a priori bounds for 2D Navier-Stokes equations (see, e.g. [28,34]), for the Boussinesq model of the Bénard convection (see [17], [14]), and also for the GOY shell model of turbulence (see [1] and [27]). Theorem 2.4 generalizes the existence result for MHD equations given in [2] to the case of multiplicative noise and also covers new situations such as the 2D magnetic Bénard problem, the 3D Leray α-model and the Sabra shell model of turbulence.…”

Section: Introductionmentioning

confidence: 82%

“…Using integration by parts, Schwarz's and Young's inequality, one checks that this map B 1 satisfies the conditions of (C1) with H = L 4 (D) 2 ∩ H (1) . The inequality in (2.3) is the well-known Ladyzhenskaya inequality (see e.g.…”

Section: Description Of the Modelmentioning

confidence: 99%

“…To see this we first note that without loss of generality we can assume that s = 1 in (2.12) (indeed, if s = 1 we can introduce a new magnetic field b := √ sb and rescale the curl of the current g := √ sg). For the velocity part of the MHD equations, we use the same spaces H (1) and V 1 and the Stokes operator generated by the bilinear form defined by (2.10) with ν = ν 1 . Now we denote this operator by A 1 .…”

Section: D Magneto-hydrodynamic Equationsmentioning

confidence: 99%

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Stochastic 2D Hydrodynamical Type Systems: Well Posedness and Large Deviations

Čhuešhov¹,

2009

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Abstract. We deal with a class of abstract nonlinear stochastic models, which covers many 2D hydrodynamical models including 2D Navier-Stokes equations, 2D MHD models and 2D magnetic Bénard problem and also some shell models of turbulence. We first prove the existence and uniqueness theorem for the class considered. Our main result is a Wentzell-Freidlin type large deviation principle for small multiplicative noise which we prove by weak convergence method.

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Section: Introductionmentioning

confidence: 82%

Section: Description Of the Modelmentioning

confidence: 99%

Section: D Magneto-hydrodynamic Equationsmentioning

confidence: 99%

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Stochastic 2D Hydrodynamical Type Systems: Well Posedness and Large Deviations

Čhuešhov¹,

2009

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“…[42] and [17]. It was shown in [14,Proposition 1] that the nonlinear term B(· , ·) for the GOY and Sabra shell models satisfies Assumption 2.2 with H = H. For more mathematical results related to shell models we refer to [3,5,6,14] and references therein.…”

Section: Goy and Sabra Shell Models Of Turbulencementioning

confidence: 99%

Irreducibility and Exponential Mixing of Some Stochastic Hydrodynamical Systems Driven by Pure Jump Noise

Fernando

Hausenblas

Razafimandimby

2016

Commun. Math. Phys.

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Abstract:In this paper, we derive several results related to the long-time behavior of a class of stochastic semilinear evolution equations in a separable Hilbert space H:Here A is a positive self-adjoint operator and B is a bilinear map, and the driving noise L is basically a D(A −1/2 )-valued Lévy process satisfying several technical assumptions. By using a density transformation theorem type for Lévy measure, we first prove a support theorem and an irreducibility property of the Ornstein-Uhlenbeck processes associated to the nonlinear stochastic problem. Second, by exploiting the previous results we establish the irreducibility of the nonlinear problem provided that for a certain γUsing a coupling argument, the exponential ergodicity is also proved under the stronger assumption that B is continuous on H × H. While the latter condition is only satisfied by the nonlinearities of GOY and Sabra shell models, the assumption under which the irreducibility property holds is verified by several hydrodynamical systems such as the 2D Navier-Stokes, Magnetohydrodynamics equations, the 3D Leray-α model, the GOY and Sabra shell models.

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“…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.…”

Section: An Example: Shell Models Of Turbulencementioning

confidence: 99%

Invariant Measures of Gaussian Type for 2D Turbulence

Bessaih

Ferrario

2012

J Stat Phys

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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).

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Some Rigorous Results on a Stochastic GOY Model

Cited by 36 publications

References 23 publications

Stochastic 2D Hydrodynamical Type Systems: Well Posedness and Large Deviations

Stochastic 2D Hydrodynamical Type Systems: Well Posedness and Large Deviations

Irreducibility and Exponential Mixing of Some Stochastic Hydrodynamical Systems Driven by Pure Jump Noise

Invariant Measures of Gaussian Type for 2D Turbulence

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