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

TL;DR

This paper establishes existence and uniqueness results for McKean-Vlasov and nonlinear Fokker-Planck equations with critical singular Lorentz kernels, including applications to 2D vorticity Navier-Stokes equations.

Contribution

It proves well-posedness for equations with critical Lorentz space drifts, extending the theory to supercritical regimes and demonstrating the nonlinear Markov property of solution laws.

Findings

Existence and conditional uniqueness of solutions in critical Lorentz spaces.

Optimality of uniqueness in dimensions d ≥ 3 with supercritical drifts.

Application to 2D vorticity Navier-Stokes equations in supercritical spaces.

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.