Ergodicity of the finite dimensional approximation of the 3D Navier-Stokes equations forced by a degenerate noise

We prove ergodicity of the finite dimensional approximations of the three dimensional Navier-Stokes equations, driven by a random force. The forcing noise acts only on a few modes and some algebraic conditions on the forced modes are found that imply the ergodicity. The convergence rate to the unique invariant measure is shown to be exponential

A simple method for the existence of a density for stochastic evolutions with rough coefficients

We extend the validity of a simple method for the existence of a density for stochastic differential equations, first introduced in [DebRom2014], by proving local estimate for the density, existence for the density with summable drift, and by improving the regularity of the density.Comment: Added a section with examples and application

H\"older regularity of the densities for the Navier--Stokes equations with noise

We prove that the densities of the finite dimensional projections of weak solutions of the Navier-Stokes equations driven by Gaussian noise are bounded and H\"older continuous, thus improving the results of Debussche and Romito [DebRom2014]. The proof is based on analytical estimates on a conditioned Fokker-Planck equation solved by the density, that has a "non-local" term that takes into account the influence of the rest of the infinite dimensional dynamics over the finite subspace under observation

Critical strong Feller regularity for Markov solutions to the Navier-Stokes equations

The main purpose of this paper is to show that Markov solutions to the 3D Navier--Stokes equations driven by Gaussian noise have the strong Feller property up to the critical topology given by the domain of the Stokes operator to the power one-fourth.Comment: submitted to JMAA for the special issue on "Stochastic PDEs in Fluid Dynamics, Particle Physics and Statistical Mechanics

An almost sure energy inequality for Markov solutions to the 3D Navier-Stokes equations

We prove existence of weak martingale solutions satisfying an almost sure version of the energy inequality and which constitute a (almost sure) Markov process.Comment: Submitted for the proceedings of the conference "Stochastic partial differential equations and applications VIII

Local existence and uniqueness in the largest critical space for a surface growth model

We show the existence and uniqueness of solutions (either local or global for small data) for an equation arising in different aspects of surface growth. Following the work of Koch and Tataru we consider spaces critical with respect to scaling and we prove our results in the largest possible critical space such that weak solutions are defined. The uniqueness of global weak solutions remains unfortunately open, unless the initial conditions are sufficiently small.Comment: 17 page

Local existence and uniqueness for a two-dimensional surface growth equation with space--time white noise

We study local existence and uniqueness for a surface growth model with space-time white noise in 2D. Unfortunately, the direct fixed-point argument for mild solutions fails here, as we do not have sufficient regularity for the stochastic forcing. Nevertheless, one can give a rigorous meaning to the stochastic PDE and show uniqueness of solutions in that setting. Using spectral Galerkin method and any other types of regularization of the noise, we obtain always the same solution