Probabilistic analysis of singularities for the 3-D Navier-Stokes equations
Probabilistic analysis of singularities for the 3D Navier-Stokes equations
The classical result on singularities for the 3D Navier-Stokes equations says that the -dimensional Hausdorff measure of the set of singular points is zero. For a stochastic version of the equation, new results are proved. For statistically stationary solutions, at any given time , with probability one the set of singular points is empty. The same result is true for a.e. initial condition with respect to a measure related to the stationary solution, and if the noise is sufficiently non degenerate...
Probabilistic models of vortex filaments
A model of vortex filaments based on stochastic processes is presented. In contrast to previous models based on semimartingales, here processes with fractal properties between and are used, which include fractional Brownian motion and similar non-Gaussian examples. Stochastic integration for these processes is employed to give a meaning to the kinetic energy.
Probability & incompressible Navier-Stokes equations: an overview of some recent developments.
Propagation of chaos for the 2D viscous vortex model
We consider a stochastic system of particles, usually called vortices in that setting, approximating the 2D Navier-Stokes equation written in vorticity. Assuming that the initial distribution of the position and circulation of the vortices has finite (partial) entropy and a finite moment of positive order, we show that the empirical measure of the particle system converges in law to the unique (under suitable a priori estimates) solution of the 2D Navier-Stokes equation. We actually prove a slightly...