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

A Central Limit Theorem for Gibbsian Invariant Measures of 2D Euler Equations

We consider Canonical Gibbsian ensembles of Euler point vortices on the 2-dimensional torus or in a bounded domain of R 2 . We prove that under the Central Limit scaling of vortices intensities, and provided that the system has zero global space average in the bounded domain case (neutrality condition), the ensemble converges to the so-called Energy-Enstrophy Gaussian random distributions. This can be interpreted as describing Gaussian fluctuations around the mean field limit of vortices ensembles. The main argument consists in proving convergence of partition functions of vortices and Gaussian distributions.Comment: 27 pages, to appear on Communications in Mathematical Physic

Stationary Solutions of Damped Stochastic 2-dimensional Euler's Equation

Existence of stationary point vortices solution to the damped and stochastically driven Euler's equation on the two dimensional torus is proved, by taking limits of solutions with finitely many vortices. A central limit scaling is used to show in a similar manner the existence of stationary solutions with white noise marginals.Comment: 24 page

Essential Self-Adjointness of Liouville Operator for 2D Euler Point Vortices

We analyse the 2-dimensional Euler point vortices dynamics in the Koopman-Von Neumann approach. Classical results provide well-posedness of this dynamics involving singular interactions for a finite number of vortices, on a full-measure set with respect to the volume measure $dx^N$ on the phase space, which is preserved by the measurable flow thanks to the Hamiltonian nature of the system. We identify a core for the generator of the one-parameter group of Koopman-Von Neumann unitaries on $L^2(dx^N)$ associated to said flow, the core being made of observables smooth outside a suitable set on which singularities can occur.Comment: 17 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

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

Nonlinear Functionals of Hyperbolic Random Waves: the Wiener Chaos Approach

We consider Gaussian random waves on hyperbolic spaces and establish variance asymptotics and central limit theorems for a large class of their integral functionals, both in the high-frequency and large domain limits. Our strategy of proof relies on a fine analysis of Wiener chaos expansions, which in turn requires us to analytically assess the fluctuations of integrals involving mixed moments of covariance kernels. Our results complement several recent findings on non-linear transforms of planar and arithmetic random waves, as well as of random spherical harmonics. In the particular case of 2-dimensional hyperbolic spaces, our analysis reveals an intriguing discrepancy between the high-frequency and large domain fluctuations of the so-called fourth polyspectra -- a phenomenon that has no counterpart in the Euclidean setting. We develop applications of a geometric flavor, most notably to excursion volumes and occupation densities.Comment: 40 pages; this is a preliminary version, comments are welcom

«Egregius formaque animisque». Un Marcello “virgiliano” in Stazio, Silvae IV 4

In silu. IV 4 Statius outlines an encomiastic portrait of Vitorius Marcellus alluding to Virgil’s description of the young Marcus Claudius Marcellus in Aen. VI. Statius not only recalls textual elements form the Aeneis, but also shapes a character coherent with Virgil’s conception of uirtus, so that Vitorius Marcellus appears to be a Virgilian hero, excellent because of his physical characteristics and aptitude for war as well as his moral qualities: egregius formaque animisque