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 $u_t-\nu \Delta u+\mathrm{div}(K(u)u)=0$ on {$(0,\infty )\times {\mathbb{R}}^2$} with Radon measures as initial data, where $K$ is the Biot–Savart operator in 2D. As a consequence, one gets the uniqueness of probabilistically weak solutions to the corresponding McKean–Vlasov stochastic differential equations (SDE). 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 ${\mathbb{P}}_{s,\zeta }$, $s\ge 0$, $\zeta$ is probability measure on ${\mathbb{R}}^d$, of path laws of the solutions to the McKean–Vlasov SDE, started with $\zeta$ at time $s$, forms a nonlinear Markov process in the sense of McKean.