Electronic Journal of Probability

Large Deviation Principle and Inviscid Shell Models

Hakima Bessaih and Annie Millet

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LDP is proved for the inviscid shell model of turbulence. As the viscosity coefficient converges to 0 and the noise intensity is multiplied by its square root, we prove that some shell models of turbulence with a multiplicative stochastic perturbation driven by a $H$-valued Brownian motion satisfy a LDP in $\mathcal{C}([0,T],V)$ 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.

Article information

Electron. J. Probab., Volume 14 (2009), paper no. 89, 2551-2579.

Accepted: 26 November 2009
First available in Project Euclid: 1 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H15: Stochastic partial differential equations [See also 35R60]
Secondary: 60F10: Large deviations 76D06: Statistical solutions of Navier-Stokes and related equations [See also 60H30, 76M35] 76M35: Stochastic analysis

Shell models of turbulence viscosity coefficient and inviscid models stochastic PDEs large deviations

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Bessaih, Hakima; Millet, Annie. Large Deviation Principle and Inviscid Shell Models. Electron. J. Probab. 14 (2009), paper no. 89, 2551--2579. doi:10.1214/EJP.v14-719. https://projecteuclid.org/euclid.ejp/1464819550

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