L^$\infty$ estimates for the jko scheme in parabolic-elliptic keller-segel systems

TL;DR

This paper establishes L^∞ estimates for densities obtained via the JKO scheme in parabolic-elliptic Keller-Segel systems, enabling short-time well-posedness results regardless of initial mass.

Contribution

It provides a general L^∞ estimate for the JKO scheme applied to Keller-Segel systems, applicable in arbitrary dimensions with arbitrary diffusion and mass.

Findings

L^∞ estimates blow up in finite time proportional to initial norm

Estimates enable short-time well-posedness for various equations

Solution existence time matches maximal Lagrangian solution time

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.