Large deviation principle and inviscid shell models. (English) Zbl 1191.60074

Bessaih, Hakima; Millet, Annie

Electron. J. Probab. 14, 2551-2579 (2009).

Summary: 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 \(C([0,T], V)\) 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.

MSC:

60H15	Stochastic partial differential equations (aspects of stochastic analysis)
60F10	Large deviations
76D06	Statistical solutions of Navier-Stokes and related equations

Keywords:

shell models of turbulence; viscosity coefficient and inviscid models; stochastic PDEs; large deviations