Global existence in a two-dimensional chemotaxis-(Navier)-Stokes system with sub-logarithmic sensitivity

Abstract

This paper considers the following chemotaxis-(Navier)-Stokes system$\{\begin{array}{cc}& n_t+u\cdot \nabla n=\Delta n-\nabla \cdot \left(n\chi \left(c\right)\nabla c\right),\\ & c_t+u\cdot \nabla c=\Delta c-f\left(n\right)c,\\ & u_t+\kappa (u\cdot \nabla )u=\Delta u+\nabla P+n\nabla \Phi \end{array}$ in a smooth bounded domain $\Omega \subset R^2$ with no-flux/no-flux/Dirichlet boundary conditions, where$\chi \left(c\right)=\frac{1}{c^{\alpha }}\text{and}f\left(n\right)=n^{\gamma }$ with $\alpha \in (0,1)$ and $\gamma \in (0,1)$. We proved that if ${\Vert c_0\Vert }_{L^{\infty }\left(\Omega \right)}<1$, then for any $\kappa \in R$ the problem possesses a global classical solution.