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

Minghua Yang; Zunwei Fu; Jinyi Sun

doi:10.3934/dcdsb.2018284

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October 2018, 23(8): 3427-3460. doi: 10.3934/dcdsb.2018284

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

Minghua Yang ^1, , Zunwei Fu ^2,3,, and Jinyi Sun ^4,

1.	Department of Mathematics, Jiangxi University of Finance and Economics, Nanchang, 330032, China
2.	Department of Mathematics, Linyi University, Linyi, 276000, China
3.	School of Mathematical Sciences, Qufu Normal University, Qufu, 273100, China
4.	Department of Mathematics, Northwest Normal University, Lanzhou, 730070, China

* Corresponding author: fuzunwei@lyu.edu.cn

Received April 2017 Revised April 2018 Published October 2018 Early access August 2018

Fund Project: 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).

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.

Keywords: Chemotaxis-Navier-Stokes equation, global solution, Besov space, Littlewood-Paley theory.

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

Citation: Minghua Yang, Zunwei Fu, Jinyi Sun. Global solutions to Chemotaxis-Navier-Stokes equations in critical Besov spaces. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3427-3460. doi: 10.3934/dcdsb.2018284

References:

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show all references

References:

[1]	M. Chae, K. Kang and J. Lee, Existence of smooth solutions to coupled chemotaxis-fuid equations, Discrete Contin. Dyn. Syst., 33 (2013), 2271-2297. doi: 10.3934/dcds.2013.33.2271. Google Scholar
[2]	M. Chae, K. Kang and J. Lee, Global existence and temporal decay in Keller-Segel models coupled to fluid equations, Comm. Partial Differ. Equ., 39 (2014), 1205-1235. doi: 10.1080/03605302.2013.852224. Google Scholar
[3]	M. Chae, K. Kang and J. Lee, Asymptotic behaviors of solutions for an aerobatic model coupled to fluid equations, J. Korean Math. Soc., 53 (2016), 127-146, arxiv.org/abs/1403.3713 doi: 10.4134/JKMS.2016.53.1.127. Google Scholar
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[35]	Y. Minsuk, L. Bataa and C. Hi, Well posedness of the Keller-Segel Navier-Stokes equations in the critical Besov spaces, Commun. Pure Appl. Anal., 14 (2015), 2453-2464. doi: 10.3934/cpaa.2015.14.2453. Google Scholar
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[37]	K. Osaki and A. Yagi, Finite dimensional attractors for one-dimensional Keller-Segel equations, Funkcial Ekvac., 44 (2001), 441-469. Google Scholar
[38]	M. Paicu, équation anisotrope de Navier-Stokes dans des espaces critiques, Rev. Mat. Iberoam., 21 (2005), 179-235. doi: 10.4171/RMI/420. Google Scholar
[39]	M. Paicu and P. Zhang, Global solutions to the 3D incompressible inhomogeneous Navier-Stokes system, J. Funct. Anal., 262 (2012), 3556-3584. doi: 10.1016/j.jfa.2012.01.022. Google Scholar
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[41]	F. Planchon, Sur un inégalité de type Poincaré, C. R. Acad. Sci. Paris Sér. I Math., 330 (2000), 21-23. doi: 10.1016/S0764-4442(00)88138-0. Google Scholar
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