Weak Solutions of McKean–Vlasov SDEs with Supercritical Drifts

Consider the following McKean–Vlasov SDE: (Formula presented.) where μX_t stands for the distribution of X_t and K(t,x):R₊×R^d→R^d is a time-dependent divergence free vector field. Under the assumption K∈L_t^q(L~_x^p) with dp+2q<2, where L~_x^p 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.