Calculus of Variations and Geometric Measure Theory

J. A. Carrillo - F. Santambrogio

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

created by santambro on 08 Sep 2017
modified on 22 Sep 2017


Accepted Paper

Inserted: 8 sep 2017
Last Updated: 22 sep 2017

Journal: Quarterly of Applied Mathematics
Year: 2017


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.