Abstract
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
{\tilde{C}}
such that the initial data
{(u_{0},n_{0},c_{0}):=(u_{0}^{h},u_{0}^{3},n_{0},c_{0})}
satisfy
\tilde{C}\bigl{(}\lVert(n_{0},c_{0})\rVert_{\dot{B}^{-2+3/q}_{q,1}(\mathbb{R}^%
{3})\times\dot{B}^{3/q}_{q,1}(\mathbb{R}^{3})}+\lVert u_{0}^{h}\rVert_{\dot{B}%
^{-1+3/p}_{p,1}(\mathbb{R}^{3})}+\lVert u_{0}^{h}\rVert_{\dot{B}^{-1+3/p}_{p,1%
}(\mathbb{R}^{3})}^{\alpha}\lVert u_{0}^{3}\rVert_{\dot{B}^{-1+3/p}_{p,1}(%
\mathbb{R}^{3})}^{1-\alpha}\bigr{)}\leq 1
for certain conditions on
{p,q}
and α implies the global existence of solutions with large initial vertical velocity component.
On the Sharp Gårding Inequality for Operators with Polynomially Bounded and Gevrey Regular Symbols
