Global Existence and Optimal Estimation of the Cauchy Problem to the 3D Fluid Equations

This paper is concerned with the Cauchy problem for a class of 3D fluid equations in an infinite cylinder. Based on Lq energy decay estimates for the corresponding linear equation searching for Green's matrix and applying Phragmen-Lindelof method, a solution space in an infinite cylinder is firstly defined. Then we prove the existence of global solutions and their optimal estimation for the 3D fluid equations with repellent and chemo-attractant. As far as we know, this is the first result about the existence of global solutions and their optimal estimation for the 3D fluid equations with repellent and chemo-attractant towards the nonlinear fluid flows in an infinite cylinder.