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

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 $\epsilon$ 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 $\epsilon$. 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 $O({\epsilon }^{1/2})$. 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.

21 pages

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