Analyticity and Existence of the Keller–Segel–Navier–Stokes Equations in Critical Besov Spaces

This paper deals with the Cauchy problem to the Keller–Segel model coupled with the incompressible 3-D Navier–Stokes equations. Based on so-called Gevrey regularity estimates, which are motivated by the works of Foias and Temam [20], we prove that the solutions are analytic for a small interval of time with values in a Gevrey class of functions. As a consequence of Gevrey estimates, we particularly imply higher-order derivatives of solutions in Besov and Lebesgue spaces. Moreover, we prove that the existence of a positive constant C~{\tilde{C}} such that the initial data (u0,n0,c0):=(u0h,u03,n0,c0){(u_{0},n_{0},c_{0}):=(u_{0}^{h},u_{0}^{3},n_{0},c_{0})} satisfy