CONVERGENCE OF THE MASS-TRANSPORT STEEPESTDESCENT SCHEME FOR THE SUBCRITICALPATLAK-KELLER-SEGEL MODEL

摘要

Variational steepest descent approximation schemes for the modified Patlak-Keller-Segel equation with a logarithmic interaction kernel in any dimension are considered. We provethe convergence of the suitably interpolated in time implicit Euler scheme, defined in terms of theEuclidean Wasserstein distance, associated with this equation for subcritical masses. As a conse-quence, we recover the recent result about the global in time existence of weak solutions to themodified Patlak-Keller- Segel equation for the logarithmic interaction kernel in any dimension in thesubcritical case. Moreover, we show how this method performs numerically in dimension one. Inthis particular case, this numerical scheme corresponds to a standard implicit Euler method for thepseudoinverse of the cumulative distribution function. We demonstrate its capabilities to reproducethe blow-up of solutions for supercritical masses easily without the need of mesh-refinement.