Large Deviations for the Stochastic Shell Model of Turbulence

In this work we first prove the existence and uniqueness of a strong solution to stochastic GOY model of turbulence with a small multiplicative noise. Then using the weak convergence approach, Laplace principle for so- lutions of the stochastic GOY model is established in certain Polish space. Thus a Wentzell-Freidlin type large deviation principle is established utilizing certain results by Varadhan and Bryc.Comment: 21 pages, submitted for publicatio

The Proper Dissipative Extensions of a Dual Pair

Let A and B be dissipative operators on a Hilbert space H and let (A,B) form a dual pair, i.e. A ? B*, resp. B ? A*. We present a method of determining the proper dissipative extensions C of this dual pair, i.e. A ? C ? B* provided that D(A) ? D(B) is dense in H. Applications to symmetric operators, symmetric operators perturbed by a relatively bounded dissipative operator and more singular differential operators are discussed. Finally, we investigate the stability of the numerical range of the different dissipative extensions

Averaging of nonautonomous damped wave equations with singularly oscillating external forces

We consider, for $\rho\in[0,1]$ and $\varepsilon>0$ small, the nonautonomous weakly damped wave equation with a singularly oscillating external force $$\partial _{t}^{2}u-\Delta u+\gamma \partial_{t}u =-f(u)+g_{0}(t)+\varepsilon ^{-\rho }g_{1}(t/\varepsilon ),$$ together with the {\it averaged} equation $$\partial _{t}^{2}u-\Delta u+\gamma \partial_{t}u =-f(u)+g_{0}(t).$$ Under suitable assumptions on the nonlinearity and the external force, we prove the uniform (w.r.t.\ $\varepsilon$) boundedness of the attractors $\mathcal{A}^\varepsilon$ in the weak energy space. If $\rho<1$, we establish the convergence of the attractor $\mathcal{A}^\varepsilon$ of the first equation to the attractor $\mathcal{A}^0$ of the second one, as $\varepsilon\to 0^+$. On the other hand, if $\rho=1$, this convergence may fail. When $\mathcal{A}^0$ is exponential, then the convergence rate of $\mathcal{A}^\varepsilon$ to $\mathcal{A}^0$ is controlled by $M\varepsilon^\eta$, for some $M\geq 0$ and some $\eta=\eta(\rho)\in(0,1)$