Stochastic Lagrangian Path for Leray’s Solutions of 3D Navier–Stokes Equations

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) ∈ R₊× R³, we show the existence of weak solutions to the following SDE, which have densities ρ_s,x(t, y) belonging to Hq1,p with p, q∈ [1 , 2) and 3p+2q>4: dXs,t=u(s,Xs,t)dt+2νdWt,Xs,s=x,t⩾s,where W is a three dimensional standard Brownian motion, ν> 0 is the viscosity constant. Moreover, we also show that for Lebesgue almost all (s, x), the solution Xs,·n(x) of the above SDE associated with the mollifying velocity field u_n weakly converges to X_s,·(x) so that X is a Markov process in almost sure sense.