McKean-Vlasov equations and nonlinear Fokker-Planck equations with critical singular Lorentz kernels

Abstract

We prove the existence and conditional uniqueness in the Krylov class for SDEs with singular divergence-free drifts in the endpoint critical Lorentz space $L^{\infty}(0,T; L^{d,\infty}(\mathbb{R}^d))$, $d \geqslant 2$, which particularly includes the $2$D Biot-Savart law. The uniqueness result is shown to be optimal in dimensions $d \geqslant 3$, by constructing different martingale solutions in the case of supercritical Lorentz drifts. As a consequence, the well-posedness of McKean-Vlasov equations and nonlinear Fokker-Planck equations with critical singular kernels is derived. In particular, this yields the uniqueness of the $2$D vorticity Navier-Stokes equations even in certain supercritical-scaling spaces. Furthermore, we prove that the path laws of solutions to McKean-Vlasov equations with critical singular kernel form a nonlinear Markov process in the sense of McKean.