Weak Solutions of McKean–Vlasov SDEs with Supercritical Drifts

Abstract

Consider the following McKean–Vlasov SDE:

$$\begin{aligned} \textrm{d} X_t=\sqrt{2}\textrm{d} W_t+\int _{{\mathbb {R}}^d}K(t,X_t-y)\mu _{X_t}(\textrm{d} y)\textrm{d} t,\ \ X_0=x, \end{aligned}$$

where $\mu _{X_t}$ stands for the distribution of $X_t$ and $K(t,x): {{\mathbb {R}}}_+\times {{\mathbb {R}}}^d\rightarrow {{\mathbb {R}}}^d$ is a time-dependent divergence free vector field. Under the assumption $K\in L^q_t({\widetilde{L}}_x^p)$ with $\frac{d}{p}+\frac{2}{q}<2$, 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.