SIAM Journal on Scientific Computing
Stability of stationary states of non-local equations with singular interaction potentials
Mathematical and Computer Modelling: An International Journal
A Note on the Subcritical Two Dimensional Keller-Segel System
Acta Applicandae Mathematicae: an international survey journal on applying mathematics and mathematical applications
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Variational steepest descent approximation schemes for the modified Patlak-Keller-Segel equation with a logarithmic interaction kernel in any dimension are considered. We prove the convergence of the suitably interpolated in time implicit Euler scheme, defined in terms of the Euclidean Wasserstein distance, associated with this equation for subcritical masses. As a consequence, we recover the recent result about the global in time existence of weak solutions to the modified Patlak-Keller-Segel equation for the logarithmic interaction kernel in any dimension in the subcritical case. Moreover, we show how this method performs numerically in dimension one. In this particular case, this numerical scheme corresponds to a standard implicit Euler method for the pseudoinverse of the cumulative distribution function. We demonstrate its capabilities to reproduce the blow-up of solutions for supercritical masses easily without the need of mesh-refinement.