McKean-Vlasov equations and nonlinear Fokker-Planck equations with critical singular Lorentz kernels
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| Format: | Preprint |
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2025
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| author | Röckner, Michael Zhang, Deng Zhao, Guohuan |
| author_facet | Röckner, Michael Zhang, Deng Zhao, Guohuan |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_13802 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | McKean-Vlasov equations and nonlinear Fokker-Planck equations with critical singular Lorentz kernels Röckner, Michael Zhang, Deng Zhao, Guohuan Probability Analysis of PDEs 39A50, 35K08, 35K67, 35Q84 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. |
| title | McKean-Vlasov equations and nonlinear Fokker-Planck equations with critical singular Lorentz kernels |
| topic | Probability Analysis of PDEs 39A50, 35K08, 35K67, 35Q84 |
| url | https://arxiv.org/abs/2505.13802 |