𝐿^{∞} estimates for the JKO scheme in parabolic-elliptic Keller-Segel systems

Carrillo, José-Antonio; Santambrogio, Filippo

doi:10.1090/qam/1493

$L^\infty$ 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
MathSciNet review: 3805040
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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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