Convergence of the mass-transport steepest descent scheme for the subcritical patlak-keller-segel model

Adrien Blanchet, Vincent Calvez, José A. Carrillo

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113 Citations (Scopus)

Abstract

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 conse-quence, 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. © 2008 Society for Industrial and Applied Mathematics.
Original languageEnglish
Pages (from-to)691-721
JournalSIAM Journal on Numerical Analysis
Volume46
DOIs
Publication statusPublished - 1 Jan 2008

Keywords

  • Patlak-Keller-Segel model
  • Steepest descent
  • Wasserstein distance

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