Markov selections for the 3D stochastic Navier-Stokes equations

We investigate the Markov property and the continuity with respect to the initial conditions (strong Feller property) for the solutions to the Navier-Stokes equations forced by an additive noise. First, we prove, by means of an abstract selection principle, that there are Markov solutions to the Navier-Stokes equations. Due to the lack of continuity of solutions in the space of finite energy, the Markov property holds almost everywhere in time. Then, depending on the regularity of the noise, we prove that any Markov solution has the strong Feller property for regular initial conditions. We give also a few consequences of these facts, together with a new sufficient condition for well-posedness.Comment: 59 pages; corrected several errors and typos, added reference

Regularity of transition semigroups associated to a 3D stochastic Navier-Stokes equation

A 3D stochastic Navier-Stokes equation with a suitable non degenerate additive noise is considered. The regularity in the initial conditions of every Markov transition kernel associated to the equation is studied by a simple direct approach. A by-product of the technique is the equivalence of all transition probabilities associated to every Markov transition kernel.Comment: 17 page

Decay of correlation rate in the mean field limit of point vortices ensembles

We consider the Mean Field limit of Gibbsian ensembles of 2-dimensional (2D) point vortices on the torus. It is a classical result that in such limit correlations functions converge to 1, that is, point vortices decorrelate: We compute the rate at which this convergence takes place by means of Gaussian integration techniques, inspired by the correspondence between the 2D Coulomb gas and the Sine-Gordon Euclidean field theory

Global regularity for a logarithmically supercritical hyperdissipative dyadic equation

We prove global existence of smooth solutions for a slightly supercritical dyadic model. We consider a generalized version of the dyadic model introduced by Katz-Pavlovic [KatPav2004] and add a viscosity term with critical exponent and a supercritical correction. This model catches for the dyadic a conjecture that for Navier-Stokes equations was formulated by Tao [Tao2009

Global regularity for a slightly supercritical hyperdissipative Navier-Stokes system

We prove global existence of smooth solutions for a slightly supercritical hyperdissipative Navier--Stokes under the optimal condition on the correction to the dissipation. This proves a conjecture formulated by Tao [Tao2009]

Cetuximab induced cutaneous rash: Does it affect psychological well-being in colorectal cancer patients?

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Analysis of equilibrium states of Markov solutions to the 3D Navier-Stokes equations driven by additive noise

We prove that every Markov solution to the three dimensional Navier-Stokes equation with periodic boundary conditions driven by additive Gaussian noise is uniquely ergodic. The convergence to the (unique) invariant measure is exponentially fast. Moreover, we give a well-posedness criterion for the equations in terms of invariant measures. We also analyse the energy balance and identify the term which ensures equality in the balance.Comment: 32 page