Weak solutions of McKean-Vlasov SDEs with supercritical drifts

TL;DR

This paper proves the existence of weak solutions for a class of McKean-Vlasov SDEs with supercritical drifts, extending to applications like the 2D-Navier-Stokes equations with measure initial data.

Contribution

It establishes weak solution existence under supercritical drift conditions and offers a new proof for 2D-Navier-Stokes solutions with measure initial vorticity.

Findings

Existence of weak solutions for McKean-Vlasov SDEs with supercritical drifts.

Application to 2D-Navier-Stokes equations with measure initial vorticity.

Extension of solution theory to less regular drift fields.

Abstract

Consider the following McKean-Vlasov SDE: $$ d X_t=\sqrt{2}d W_t+\int_{{\mathbb R}^d}K(t,X_t-y)\mu_{X_t}(dy)d t,\ \ X_0=x, $$ where $\mu_{X_t}$ stands for the distribution of $X_t$ and $K(t,x): {\mathbb R}_+\times{\mathbb R}^d\to{\mathbb R}^d$ is a time-dependent divergence free vector field. Under the assumption $K\in L^q_t(\widetilde L_x^p)$ with $\frac dp+\frac2q<2$, where $\widetilde L^p_x$ stands for the localized $L^p$-space, we show the existence of weak solutions to the above SDE. As an application, we provide a new proof for the existence of weak solutions to 2D-Navier-Stokes equations with measure as initial vorticity.