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

  • * Corresponding author: fuzunwei@lyu.edu.cn

    * Corresponding author: fuzunwei@lyu.edu.cn 
This paper was partially supported by the National Natural Science Foundation of China (Grant Nos. 11671185, 11771195), the Postdoctoral Science Foundation of Jiangxi Province (Grant No. 2017KY23) and Educational Commission Science Programm of Jiangxi Province (Grant No. GJJ170345)
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  • 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.

    Mathematics Subject Classification: Primary: 35K55, 35Q35, 92C17; Secondary: 42B35.


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