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

TL;DR

This paper proves the uniqueness of distributional solutions to the 2D vorticity Navier-Stokes equations with Radon measure initial data, establishes stochastic representations, and demonstrates the associated nonlinear Markov process structure.

Contribution

It establishes the uniqueness of solutions in the vorticity form, links these solutions to McKean-Vlasov SDEs, and shows the nonlinear Markov process structure of their laws.

Findings

Uniqueness of distributional solutions for 2D vorticity Navier-Stokes.

Existence of stochastic representations via McKean-Vlasov SDEs.

Pathwise uniqueness and strong solutions for initial data with $L^4$ density.

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 $L^{\frac43}$ 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 $\mathbb{P}_{s,\zeta},$ $s \geq 0$, $\zeta=$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.