A unified relative entropy framework for macroscopic limits of Vlasov--Fokker--Planck equations

Abstract

We develop a unified relative entropy framework for macroscopic limits of kinetic equations with Riesz-type interactions and Fokker-Planck relaxation. Our analysis covers three prototypical singular regimes: the diffusive limit leading to a drift-diffusion equation, the high-field limit yielding the aggregation equation in the repulsive regime, and the strong magnetic field limit producing a generalized surface quasi-geostrophic equation. The method combines entropy dissipation, Fisher-information control, and modulated interaction energies into a robust stability theory yielding both strong and weak convergence results. For the strong convergence, we establish quantitative relative entropy estimates toward macroscopic limits under well-prepared initial data, extending the scope of the method to settings where nonlocal forces and singular scalings play a decisive role. For the weak convergence, our approach captures three complementary phenomena: in the diffusive regime, it yields sharper quantitative estimates in weak topologies consistent with the formally optimal scaling; in the high-field regime, it propagates bounded Lipschitz stability for a class of mildly prepared initial data, even when the relative entropy diverges with respect to the singular scaling parameter; and in the strong magnetic field regime, it provides quantitative weak estimates, including bounded Lipschitz control of the rescaled momentum and negative Sobolev control of the density. This broader perspective shows that relative entropy provides not only a tool for strong convergence, but also a mechanism for treating low-regularity and mildly prepared regimes. The analysis highlights the unifying role of relative entropy in connecting microscopic dissipation with both strong and weak macroscopic convergence.