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| We consider some shell models of turbulence in a very general form. These are phenomenological approximations of the Navier-Stokes equations, with a viscous linear part that is dissipative and a nonlinear part that is not globally Lipschitz. We assume that this model is driven by a multiplicative
noise (fBM with Hurst parameter $H>1/2$). We will assume that the nontrivial diffusion term which is a nonlinear operator will satisfy some Lipschitz property and some other differentiability conditions.
We will prove the existence and uniqueness of a global variational solution. The proof will be achieved in two steps. The first step is to prove that the variational solutions exist and are unique for a smooth noise. Moreover, some a priori estimates will be obtained in some functional spaces. The second step uses a compactness argument. In fact, we prove that these solutions have a limit when the smooth noise converges to the fBM and that the limit is a variational solution for the shell model. All these statements and proofs are based on a pathwise argument.
We hope to extend these results to the 2d Navier-Stokes equations. |
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