American Institute of Mathematical Sciences

The paper deals with the Cauchy problem of Navier-Stokes-Nernst-Planck-Poisson system (NSNPP). First of all, based on so-called Gevrey regularity estimates, which is motivated by the works of Foias and Temam [J. Funct. Anal., 87 (1989), 359-369], we prove that the solutions are analytic in a Gevrey class of functions. As a consequence of Gevrey estimates, we particularly obtain higher-order derivatives of solutions in Besov and Lebesgue spaces. Finally, we prove that there exists a positive constant

$\mathbb{C}$

such that if the initial data

$(u_{0}, n_{0}, c_{0})=(u_{0}^{h}, u_{0}^{3}, n_{0}, c_{0})$

satisfies

$\begin{aligned}&\|(n_{0}, c_{0},u_{0}^{h})\|_{\dot{B}^{-2+3/q}_{q, 1}× \dot{B}^{-2+3/q}_{q, 1}×\dot{B}^{-1+3/p}_{p, 1}}+\|u_{0}^{h}\|_{\dot{B}^{-1+3/p}_{p, 1}}^{α}\|u_{0}^{3}\|_{\dot{B}^{-1+3/p}_{p, 1}}^{1-α}≤q1/\mathbb{C}\end{aligned}$

for

$p, q, α$

with

$1<p<q≤ 2p<\infty, \frac{1}{p}+\frac{1}{q}>\frac{1}{3}, 1< q<6, \frac{1}{p}-\frac{1}{q}≤\frac{1}{3}$

, then global existence of solutions with large initial vertical velocity component is established.

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.