Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 
 

 

$ L^\infty$ estimates for the JKO scheme in parabolic-elliptic Keller-Segel systems


Authors: José-Antonio Carrillo and Filippo Santambrogio
Journal: Quart. Appl. Math. 76 (2018), 515-530
MSC (2010): Primary 35K55; Secondary 49K20
DOI: https://doi.org/10.1090/qam/1493
Published electronically: November 7, 2017
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Abstract: We prove $ L^\infty $ estimates on the densities that are obtained via the JKO scheme for a general form of a parabolic-elliptic Keller-Segel type system, with arbitrary diffusion, arbitrary mass, and in arbitrary dimension. Of course, such an estimate blows up in finite time, a time proportional to the inverse of the initial $ L^\infty $ norm. This estimate can be used to prove short-time well-posedness for a number of equations of this form regardless of the mass of the initial data. The time of existence of the constructed solutions coincides with the maximal time of existence of Lagrangian solutions without the diffusive term by characteristic methods.


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José-Antonio Carrillo
Affiliation: Department of Mathematics, Imperial College London, South Kensington Campus, SW7 2AZ London, United Kingdom
Email: carrillo@imperial.ac.uk

Filippo Santambrogio
Affiliation: Laboratoire de Mathématiques d’Orsay, Univ. Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay Cedex, France
Email: filippo.santambrogio@math.u-psud.fr

DOI: https://doi.org/10.1090/qam/1493
Received by editor(s): September 10, 2017
Published electronically: November 7, 2017
Additional Notes: The first author was partially supported by the Royal Society via a Wolfson Research Merit Award and by EPSRC grant number EP/P031587/1.
The work was finished during a visit of the second author to the Imperial College, in the framework of a joint CNRS-Imperial Fellowship; the hospitality and the financial support of the Imperial College are warmly acknowledged.
Article copyright: © Copyright 2017 Brown University

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