Gaussian invariant measures and stationary solutions of 2D Primitive Equations

We introduce a Gaussian measure formally preserved by the 2-dimensional Primitive Equations driven by additive Gaussian noise. Under such measure the stochastic equations under consideration are singular: we propose a solution theory based on the techniques developed by Gubinelli and Jara in \cite{GuJa13} for a hyperviscous version of the equations.Comment: 15 page

Quantitative mixing and dissipation enhancement property of Ornstein-Uhlenbeck flow

This work deals with mixing and dissipation ehancement for the solution of advection-diffusion equation driven by a Ornstein-Uhlenbeck velocity field. We are able to prove a quantitative mixing result, uniform in the diffusion parameter, and enhancement of dissipation over a finite time horizon.Comment: 25 page

Large Deviations for Stochastic equations in Hilbert Spaces with non-Lipschitz drift

We prove a Freidlin-Wentzell result for stochastic differential equations in infinite-dimensional Hilbert spaces perturbed by a cylindrical Wiener process. We do not assume the drift to be Lipschitz continuous, but only continuous with at most linear growth. Our result applies, in particular, to a large class of nonlinear fractional diffusion equations perturbed by a space-time white noise.Comment: 16 page

Equilibrium Statistical Mechanics of Barotropic Quasi-Geostrophic Equations

We consider equations describing a barotropic inviscid flow in a channel with topography effects and beta-plane approximation of Coriolis force, in which a large-scale mean flow interacts with smaller scales. Gibbsian measures associated to the first integrals energy and enstrophy are Gaussian measures supported by distributional spaces. We define a suitable weak formulation for barotropic equations, and prove existence of a stationary solution preserving Gibbsian measures, thus providing a rigorous infinite-dimensional framework for the equilibrium statistical mechanics of the model.Comment: 18 page

Burst of Point Vortices and Non-Uniqueness of 2D Euler Equations

We give a rigorous construction of solutions to the Euler point vortices system in which three vortices burst out of a single one in a configuration of many vortices, or equivalently that there exist configurations of arbitrarily many vortices in which three of them collapse in finite time. As an intermediate step, we show that well-known self-similar bursts and collapses of three isolated vortices in the plane persist under a sufficiently regular external perturbation. We also discuss how our results produce examples of non-unique weak solutions to 2-dimensional Euler's equations -- in the sense introduced by Schochet -- in which energy is dissipated.Comment: 30 page

2D Euler equations with Stratonovich transport noise as a large scale stochastic model reduction

The limit from an Euler type system to the 2D Euler equations with Stratonovich transport noise is investigated. A weak convergence result for the vorticity field and a strong convergence result for the velocity field are proved. Our results aim to provide a stochastic reduction of fluid-dynamics models with three different time scales.Comment: 30 page

Kolmogorov 4/54/5 law for the forced 3D Navier-Stokes equations

We prove that the solutions to the 3D forced Navier-Stokes equations constructed by Bru\`e, Colombo, Crippa, De Lellis and Sorella satisfy an $L^p$-in-time version of the Kolmogorov 4/5 law for behavior of the averaged third order longitudinal structure function along the vanishing viscosity limit. The result has a natural probabilistic interpretation: the predicted behavior is observed on average after waiting for some sufficiently generic random time. This is then applied to derive a bound for the exponent of the third order absolute structure function in accordance with the Kolmogorov turbulence theory

Stochastic model reduction : convergence and applications to climate equations

We study stochastic model reduction for evolution equations in infinite-dimensional Hilbert spaces and show the convergence to the reduced equations via abstract results of Wong–Zakai type for stochastic equations driven by a scaled Ornstein–Uhlenbeck process. Both weak and strong convergence are investigated, depending on the presence of quadratic interactions between reduced variables and driving noise. Finally, we are able to apply our results to a class of equations used in climate modeling

Large Deviations for SDEs

Large Deviations concern about giving sharp logarithmic asymptotics as $\varepsilon \to 0$ for the probabilities $\mu^\varepsilon(A)$, where $\mu^\varepsilon$ is a family of probability measures on a metric space indexed by $\varepsilon>0$. We consider mostly the case where $\mu^\varepsilon$ is the law of a random process, solution to certain SDE, with noise intensity equal to $\varepsilon$. Starting from the classical Friedlin-Wentzell Theorem, that treats the case of drift $b$ bounded and Lipschitz continuous ,we find weaker sufficient conditions on $b$ which guarantee the validity of a Large Deviations Principle. Moreover, in the particular case $b(x) = x \|x\|^{\gamma-1}$, $\gamma \in (0,1)$, in addition to the previous one we establish a second Large Deviation Principle, strictly related to the Peano phenomenon