Formulation of Chimera Gradient Flows for Chemotaxis Systems with Indirect Signal Production and Degenerate Diffusion

TL;DR

This paper develops a novel formulation of chimera gradient flows for chemotaxis systems with indirect signal production and degenerate diffusion, establishing existence of global solutions via a Lyapunov functional.

Contribution

It introduces a new approach to treat chemotaxis systems as coupled gradient flows and proves the existence of global solutions using a Lyapunov functional.

Findings

Existence of time-global solutions for the chemotaxis system.

Construction of approximate solutions via a modified minimizing movement scheme.

Identification of a Lyapunov functional ensuring compactness and solution existence.

Abstract

A parabolic system of three unknown functions, not expressible as gradient flows, is treated as three coupled gradient flows. For each unknown function, the minimizing movement scheme is used to construct a time-discrete approximate solution. Unlike standard minimizing movement scheme for gradient flows, the relative compactness of the time-discrete approximate solution with respect to the time step is not inherently guaranteed. However, the existence of a Lyapunov functional ensures this relative compactness, leading to the existence of time-global solutions.