Quantitative Propagation of Chaos for 2D Viscous Vortex Model with General Circulations on the Whole Space

Abstract: We derive the quantitative propagation of chaos in the sense of relative entropy for the 2D viscous vortex model with general circulations, approximating the vorticity formulation of the 2D Navier-Stokes equation on the whole Euclidean space, which is an extension of the previous works \cite{fournier2014propagation,jabin2018quantitative}. We improve our previous results in \cite{feng2023quantitative} to the more general case that the vorticity may change sign, allowing the circulations to be in different orientations. We also extend the results of \cite{wang2024sharp} to the whole space to obtain the sharp local propagation of chaos in the high viscosity regime. Both results of convergence rates are global-in-time on any finite time horizon and optimal among existing literature, thanks to the careful estimates using the Grigor'yan parabolic maximum principle and a new ODE hierarchy and iterated integral estimates.