Global solutions to Chemotaxis-Navier-Stokes equations in critical Besov spaces

In this article, we consider the Cauchy problem to chemotaxis model coupled to the incompressible Navier-Stokes equations. Using the Fourier frequency localization and the Bony paraproduct decomposition, we establish the global-in-time existence of the solution when the gravitational potential ϕ and the small initial data $(u_{0}, n_{0}, c_{0})$ in critical Besov spaces under certain conditions. Moreover, we prove that there exist two positive constants σ₀ and $C_{0}$ such that if the gravitational potential $\phi \in \dot B_{p,1}^{3/p}({\mathbb{R}^3})$ and the initial data $(u_{0}, n_{0}, c_{0}): = (u_{0}^{1}, u_{0}^{2}, u_{0}^{3}, n_{0}, c_{0}): = (u_{0}^{h}, u_{0}^{3}, n_{0}, c_{0})$ satisfies

$\begin{equation*}\begin{aligned} &\left(\left\|u_{0}^{h}\right\|_{\dot{B}^{-1+3/p}_{p, 1}(\mathbb{R}^3)}+\left\|\left(n_{0}, c_{0}\right)\right\|_{\dot{B}^{-2+3/q}_{q, 1}(\mathbb{R}^3) \times \dot{B}^{3/q}_{q, 1}(\mathbb{R}^3)}\right)\\ &\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \times\exp\left\{C_{0}\left(\left\|u_{0}^{3}\right\|_{\dot{B}^{-1+3/p}_{p, 1}(\mathbb{R}^3)}+1\right)^{2}\right\} \leq \sigma_{0}\end{aligned}\end{equation*}$

for some $p, q$ with $1<p, q<6,\frac{1}{p}+\frac{1}{q}>\frac{2}{3}$ and $\frac{1}{\min\{p, q\}}-\frac{1}{\max\{p, q\}} \le \frac{1}{3}$, then the global existence results can be extended to the global solutions without any small conditions imposed on the third component of the initial velocity field $u_{0}^{3}$ in critical Besov spaces with the aid of continuity argument. Our initial data class is larger than that of some known results. Our results are completely new even for three-dimensional chemotaxis-Navier-Stokes system.