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

2025/05/20 by Michael Röckner, Deng Zhang, Röckner, Michael +3
Economics, Econometrics and Finance · Mathematics · #35K08 #35K67 #35Q84 #39A50 #Analysis of PDEs (math.AP) #FOS: Mathematics #Gas Dynamics and Kinetic Theory #Navier-Stokes equation solutions #Probability (math.PR) #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.2505.13802

openalex publication_date 2025/05/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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^∞(0,T; L^d,∞(ℝ^d)), d \geqslant 2, which particularly includes the 2D 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 2D 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.

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