Measure valued solutions of the 2D Keller-Segel system

TL;DR

This paper investigates measure-valued solutions of the 2D Keller-Segel system, providing a framework for understanding solutions as limits of regularized systems and exploring their dependence on regularization methods.

Contribution

It introduces a generalized measure-valued solution concept for the Keller-Segel system and analyzes how these solutions depend on different regularization approaches.

Findings

Existence of measure-valued solutions as limits of regularizations

Dependence of solutions on the regularization method

Extension of solution concepts to include blow-up behavior

Abstract

We study the solutions of the two-dimensional Keller-Segel system describing chemotaxis. The Keller-Segel system as well as the properties of the blow-up set has been extensively studied. In this paper we obtain generalized solutions for the Keller-Segel system in the sense of measures as the limit of two-different regularizations of it. We will show that the resulting limit measures, that are in some suitable sense global weak solutions of the Keller-Segel system, depend in the regularization.