Weak Solutions to the Three-Dimensional Chemotaxis-Stokes System with Active Transport

In this paper, we study the following chemotaxis-Stokes system with active transport <img width="280" height="70" src="http://article.sciencepublishinggroup.com/journal/322/1811490/image001.png" /> in a bounded domain with positive parameters <i>α</i>, <i>β</i>, <i>χ</i> and <i>λ</i>. Here, <i>c</i> and <i>n</i> denote the density of the nutrient acting as a chemoattractant and the density of the cell, respectively and <i>u</i> denotes the velocity of the fluid. The parameters <i>α</i>, <i>β</i>, <i>χ</i> and <i>λ</i> are positive constants and <i>λ</i> represents the cell growth rate occurred by the nutrient supply. The novelty of this system is that there is not only a chemotaxis term, which reflects the movement of the cells toward nutrient sources, but also an active transport one, which means the nutrient is moving towards the cells. In order to prove the existence of weak solutions to the above problem, we introduce the regularized problem using the the Yosida approximation of <i>B</i> := −∆ + 1 under homogeneous Neumann boundary conditions in <i>L</i><sup>2</sup> (Ω). Based on a priori estimates for the solutions of the regularized problem, it is shown that under no-flux boundary condition and for any suitably regular initial data, an associated initial value problem possesses a global weak solution provided 0 < <i>χ</i> < 1.