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arXiv · 1503.07268

Hölder regularity and Uniqueness theorem on weak solutions to the degenerate Keller-Segel system

Sunghoon Kim·Ki-Ahm Lee
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Abstract

In this paper, we present local Hölder estimates for the degenerate Keller-Segel system \eqref{eq-cases-aligned-main-problem-of-Keller-Segel-System} below in the range of $m>1$ and $q>1$ before a blow-up of solutions. To deal with difficulties caused by the degeneracy of the operator, we find uniform estimates depending sup-norm of the density function and modified the energy estimates and intrinsic scales considered in Porous Medium Equation. As its application, the uniqueness of weak solution to \eqref{eq-cases-aligned-main-problem-of-Keller-Segel-System} is also showed in the class of Hölder continuous functions by proving $L^1$-contraction in this class.

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Sunghoon Kim, Ki-Ahm Lee. 2015-03-25. Hölder regularity and Uniqueness theorem on weak solutions to the degenerate Keller-Segel system. https://arxiv.org/abs/1503.07268

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