Inviscid limit of stochastic damped 2D Navier-Stokes equations

We consider the inviscid limit of the stochastic damped 2D Navier- Stokes equations. We prove that, when the viscosity vanishes, the stationary solution of the stochastic damped Navier-Stokes equations converges to a stationary solution of the stochastic damped Euler equation and that the rate of dissipation of enstrophy converges to zero. In particular, this limit obeys an enstrophy balance. The rates are computed with respect to a limit measure of the unique invariant measure of the stochastic damped Navier-Stokes equations

Uniqueness of solutions of the stochastic Navier-Stokes equation with invariant measure given by the enstrophy

A stochastic Navier-Stokes equation with space-time Gaussian white noise is considered, having as infinitesimal invariant measure a Gaussian measure \mu_{\nu} whose covariance is given in terms of the enstrophy. Pathwise uniqueness for \mu_{\nu}-a.e. initial velocity is proven for solutions having \mu_{\nu} as invariant measure.Comment: Published by the Institute of Mathematical Statistics (http://www.imstat.org) in the Annals of Probability (http://www.imstat.org/aop/) at http://dx.doi.org/10.1214/00911790400000037

A note on a result of Liptser-Shiryaev

Given two stochastic equations with different drift terms, under very weak assumptions Liptser and Shiryaev provide the equivalence of the laws of the solutions to these equations by means of Girsanov transform. Their assumptions involve both the drift terms. We are interested in the same result but with the main assumption involving only the difference of the drift terms. Applications of our result will be presented in the finite as well as in the infinite dimensional setting.Comment: 22 pages; revised and enlarged versio

Lp-solutions of the Navier-Stokes equation with fractional Brownian noise

We study the Navier-Stokes equations on a smooth bounded domain D ⊂ Rd (d = 2 or 3), under the effect of an additive fractional Brownian noise. We show local existence and uniqueness of a mild Lp-solution for p > d.34539553FUNDAÇÃO DE AMPARO À PESQUISA DO ESTADO DE SÃO PAULO - FAPESP2017/17670-0; 2015/07278-

Statistical properties of stochastic 2D Navier-Stokes equations from linear models

A new approach to the old-standing problem of the anomaly of the scaling exponents of nonlinear models of turbulence has been proposed and tested through numerical simulations. This is achieved by constructing, for any given nonlinear model, a linear model of passive advection of an auxiliary field whose anomalous scaling exponents are the same as the scaling exponents of the nonlinear problem. In this paper, we investigate this conjecture for the 2D Navier-Stokes equations driven by an additive noise. In order to check this conjecture, we analyze the coupled system Navier-Stokes/linear advection system in the unknowns $(u,w)$. We introduce a parameter $\lambda$ which gives a system $(u^\lambda,w^\lambda)$; this system is studied for any $\lambda$ proving its well posedness and the uniqueness of its invariant measure $\mu^\lambda$. The key point is that for any $\lambda \neq 0$ the fields $u^\lambda$ and $w^\lambda$ have the same scaling exponents, by assuming universality of the scaling exponents to the force. In order to prove the same for the original fields $u$ and $w$, we investigate the limit as $\lambda \to 0$, proving that $\mu^\lambda$ weakly converges to $\mu^0$, where $\mu^0$ is the only invariant measure for the joint system for $(u,w)$ when $\lambda=0$.Comment: 23 pages; improved versio