Global well-posedness and zero-diffusion limit of classical solutions to the 3D conservation laws arising in chemotaxis

TL;DR

This paper proves the global existence of classical solutions and analyzes the zero-diffusion limit for 3D chemotaxis models derived from Keller-Segel, establishing convergence rates and improving well-posedness results.

Contribution

It establishes global well-posedness for diffusive and non-diffusive chemotaxis models and quantifies the convergence rate as diffusion vanishes.

Findings

Global well-posedness of classical solutions for diffusive model

Convergence rate of order O(ε^{1/2}) in L^ norm

Improved well-posedness results for the non-diffusive model

Abstract

In this paper, we study the relationship between a diffusive model and a non-diffusive model which are both derived from the well-known Keller-Segel model, as a coefficient of diffusion $\varepsilon$ goes to zero. First, we establish the global well-posedness of classical solutions to the Cauchy problem for the diffusive model with smooth initial data which is of small $L^2$ norm, together with some {\it a priori} estimates uniform for $t$ and $\varepsilon$. Then we investigate the zero-diffusion limit, and get the global well-posedness of classical solutions to the Cauchy problem for the non-diffusive model. Finally, we derive the convergence rate of the diffusive model toward the non-diffusive model. It is shown that the convergence rate in $L^\infty$ norm is of the order $O(\varepsilon^{1/2})$. It should be noted that the initial data is small in $L^2$-norm but can be of large oscillations with constant state at far field. As a byproduct, we improve the corresponding result on the well-posedness of the non-difussive model which requires small oscillations.