PUB - Publikationen an der Universität Bielefeld

In this paper we show the existence of stochastic Lagrangian particle trajectory for Leray's solution of 3D Navier-Stokes equations. More precisely, for any Leray's solution u of 3D-NSE and each (s, x) is an element of R+ x R-3, we show the existence of weak solutions to the following SDE, which have densities rho(s, x) (t, y) belonging to H-q(1, p) with p, q is an element of [1, 2) and 3/p + 2/q > 4: dX(s, t) = u(s, X-s,X- t)dt + root 2 nu dW(t), X-s,X- s = x, t >= s, where W is a three dimensional standard Brownian motion, nu > 0 is the viscosity constant. Moreover, we also showthat for Lebesgue almost all (s, x), the solution X-s(n),. (x) of the above SDE associated with the mollifying velocity field un weakly converges to X-s, . ( x) so that X is a Markov process in almost sure sense.