Abstract
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.
Highlights
Of W, as well as apriori bounds of the solution in C([0, T ], V ) when the initial condition belong to V and under reinforced assumptions on σ
In order to define the stochastic control equation, we introduce for ν ≥ 0 a family of intensity coefficients σν of a random element h ∈ AM for some M > 0
We prove the main result of this section
Summary
3 is mainly devoted to prove existence, uniqueness of the solution to the deterministic inviscid equation with an external multiplicative impulse driven by an element of the RKHS of W , as well as apriori bounds of the solution in C([0, T ], V ) when the initial condition belong to V and under reinforced assumptions on σ. Under these extra assumptions, we are able to improve the apriori estimates of the solution and establish them in C([0, T ], V ) and L2([0, T ], Dom(A)).
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