Uniqueness of distributional solutions to the 2D vorticity Navier-Stokes equation and its associated nonlinear Markov process

Abstract

In this work we prove uniqueness of distributional solutions to $2D$ Navier-Stokes equations in vorticity form $u_t-\nu \Delta u+div(K(u)u)=0$ on $(0,\infty )\times {\mathbb{R}}^2$ with Radon measures as initial data, where $K$ is the Biot-Savart operator in 2-D. As a consequence, one gets the uniqueness of probabilistically weak solutions to the corresponding McKean-Vlasov stochastic differential equations. It is also proved that for initial conditions with density in $L^4$ these solutions are strong, so can be written as a functional of the Wiener process, and that pathwise uniqueness holds in the class of weak solutions, whose time marginal law densities are in $L^{\frac{4}{3}}$ in space-time. In particular, one derives a stochastic representation of the vorticity $u$ of the fluid flow in terms of a solution to the McKean-Vlasov SDE. Finally, it is proved that the family ${\mathbb{P}}_{s,\zeta },$ $s\ge 0$, $\zeta =$probability measure on ${\mathbb{R}}^d$, of path laws of the solutions to the McKean-Vlasov SDE, started with $\zeta$ at $s$, form a nonlinear Markov process in the sense of McKean.