Global well-posedness for the 2D chemotaxis-Euler equations modeling coral fertilization

Abstract

In this paper, we study the Cauchy problem for a chemotaxis-Euler equations modeling coral fertilization in $\mathbb{R}^{2}$. We first establish the local well-posedness of this model in Sobolev space. In the latter, motivated by the paper \cite{6,18}, by using the $W^{1,p}$-energy estimate ($2<p<\infty$) and the Br\'{e}zis-Wainger inequality, we can extend these local solutions to global ones under the small initial data assumption.