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 and Continuous Dynamical Systems - B, 2018, 23 (8) : 3427-3460. doi: 10.3934/dcdsb.2018284

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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.
[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.
[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.
[4]	Y. Chemin, Théorémes dunicité pour le systéme de Navier-Stokes tridimensionnal, J. Anal. Math., 77 (1999), 27-50. doi: 10.1007/BF02791256.
[5]	Y. Chemin, M. Paicu and P. Zhang, Global large solutions to 3-D inhomogeneous Navier-Stokes system with one slow variable diffusion, J. Differential Equations, 256 (2014), 223-252. doi: 10.1016/j.jde.2013.09.004.
[6]	Y. Chemin and P. Zhang, On the global well-posedness to the 3D incompressible anisotropic Navier-Stokes equations, Comm. Math. Phys., 272 (2007), 529-566. doi: 10.1007/s00220-007-0236-0.
[7]	H. Choe and B. Lkhagvasuren, Global existence result for Chemotaxis Navier-Stokes equations in the critical Besov spaces, J. Math. Anal. Appl., 446 (2017), 1415-1426. doi: 10.1016/j.jmaa.2016.09.050.
[8]	Y. Chung and K. Kang, Existence of global solutions for a Keller-Segel-fluid equations with nonlinear diffusion, arXiv: 1504.02274.
[9]	P. Constantin, A. Kiselev, L. Ryzhik and A. Zlatoš, Diffusion and mixing in fluid flow, Ann. Math., 168 (2008), 643-674. doi: 10.4007/annals.2008.168.643.
[10]	R. Danchin, Fourier Analysis Methods for PDEs, Lecture Notes, 14, November 2005.
[11]	R. Danchin, Local theory in critical spaces for compressible viscous and heat-conducting gases, Comm. Partial Differ. Equ., 26 (2001), 1183-1233. doi: 10.1081/PDE-100106132.
[12]	R. Duan, A. Lorz and P. Markowich, Global solutions to the coupled chemotaxis-fuid equations, Comm. Partial Differ. Equ., 35 (2010), 1635-1673. doi: 10.1080/03605302.2010.497199.
[13]	R. Duan and Z. Xiang, Global solutions to the coupled chemotaxis-fuid equations, Int. Math. Res. Not. IMRN, (2014), 1833-1852. doi: 10.1093/imrn/rns270.
[14]	E. Espejo and T. Suzuki, Reaction terms avoiding aggregation in slow fluids, Nonlinear Anal. Real World Appl., 21 (2015), 110-126. doi: 10.1016/j.nonrwa.2014.07.001.
[15]	M. Francesco, A. Lorz and P. Markowich, Chemotaxis-fluid coupled model for swimming bacteria with nonlinear diffusion: global existence and asymptotic behavior, Discrete Contin. Dyn. Syst., 28 (2010), 1437-1453. doi: 10.3934/dcds.2010.28.1437.
[16]	H. Fujita and T. Kato, On the Navier-Stokes initial value problem I, Arch. Ration. Mech. Anal., 16 (1964), 269-315. doi: 10.1007/BF00276188.
[17]	B. Hajer, Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Springer, Berlin, 2011. doi: 10.1007/978-3-642-16830-7.
[18]	M. Herrero and L. Velazquez, A blow-up mechanism for chemotaxis model, Ann. Sc. Norm. Super. Pisa., 24 (1997), 633-683.
[19]	D. Horstman and G. Wang, Blow-up in a chemotaxis model without symmetry assumptions, European J. Appl. Math., 12 (2001), 159-177. doi: 10.1017/S0956792501004363.
[20]	J. Huang, M. Paicu and P. Zhang, Global well-posedness of incompressible inhomogeneous fluid systems with bounded density or non-Lipschitz velocity, Arch. Ration. Mech. Anal., 209 (2013), 631-382. doi: 10.1007/s00205-013-0624-x.
[21]	T. Kato, Strong L^p-solutions of the Navier-Stokes equation in $\mathbb{R}^m$, with applications to weak solutions, Math. Z., 187 (1984), 471-480. doi: 10.1007/BF01174182.
[22]	E. Keller and L. Segel, Initiation of slide mold aggregation viewd as an instability, J. Theor. Biol., 6 (1970), 399-415.
[23]	E. Keller and L. Segel, Model for chemotaxis, J. Theor. Biol., 30 (1971), 225-234. doi: 10.1016/0022-5193(71)90050-6.
[24]	A. Kiselev and L. Ryzhik, Enhancement of the traveling front speeds in reaction-diffusion equations with advection, Annales de l'Institut Henri Poincaré(C) Non Linear Analysis, 18 (2001), 309-358. doi: 10.1016/S0294-1449(01)00068-3.
[25]	A. Kiselev and X. Xu, Suppression of chemotactic explosion by mixing, Arch. Ration. Mech. Anal., 222 (2016), 1077-1112. doi: 10.1007/s00205-016-1017-8.
[26]	J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248. doi: 10.1007/BF02547354.
[27]	H. Koch and D. Tataru, Well-posedness for the Navier-Stokes equations, Adv. Math., 157 (2001), 22-35. doi: 10.1006/aima.2000.1937.
[28]	H. Kozono and M. Nakao, Periodic solutions of the Navier-Stokes equations in unbounded domains, Tohoku Math. J., 48 (1996), 33-50. doi: 10.2748/tmj/1178225411.
[29]	H. Kozono and M. Yamazaki, Semilinear heat equations and the Navier-Stokes equation with distributions in new function spaces as initial data, Comm. Partial Differ. Equ., 19 (1994), 959-1014. doi: 10.1080/03605309408821042.
[30]	H. Liu and H. Gao, Global well-posedness and long time decay of the 3D Boussinesq equations, J. Differential Equations, 263 (2017), 8649-8665. doi: 10.1016/j.jde.2017.08.049.
[31]	J. Liu and A. Lorz, A coupled chemotaxis-fluid model: Global existence, Ann. Inst. H.Poincaré Anal. Non Linéaire, 28 (2011), 643-652. doi: 10.1016/j.anihpc.2011.04.005.
[32]	Q. Liu, T. Zhang and J. Zhao, Global solutions to the 3D incompressible nematic liquid crystal system, J. Differential Equations, 258 (2015), 1519-1547. doi: 10.1016/j.jde.2014.11.002.
[33]	A. Lorz, Coupled chemotaxis fuid model, Math. Models Methods Appl. Sci., 20 (2010), 987-1004. doi: 10.1142/S0218202510004507.
[34]	A. Lorz, A coupled Keller-Segel-Stokes model: Global existence for small initial data and blow-up delay, Commun. Math. Sci., 10 (2012), 555-574. doi: 10.4310/CMS.2012.v10.n2.a7.
[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.
[36]	T. Nagai, T. Senba and K. Yoshida, Applications of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial Ekvac., 40 (1997), 411-433.
[37]	K. Osaki and A. Yagi, Finite dimensional attractors for one-dimensional Keller-Segel equations, Funkcial Ekvac., 44 (2001), 441-469.
[38]	M. Paicu, équation anisotrope de Navier-Stokes dans des espaces critiques, Rev. Mat. Iberoam., 21 (2005), 179-235. doi: 10.4171/RMI/420.
[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.
[40]	C. Patlak, Random walk with persistence and external bias, Bull. Math. Biol. Biophys., 15 (1953), 311-338. doi: 10.1007/BF02476407.
[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.
[42]	Y. Tao, Boundedness in a chemotaxis model with oxygen consumption by bacteria, J. Math. Anal. Appl., 381 (2011), 521-529. doi: 10.1016/j.jmaa.2011.02.041.
[43]	Y. Tao and M. Winkler, Global existence and boundedness in a Keller-Segel-Stokes model with arbitrary porous medium diffusion, Discrete Contin. Dyn. Syst., 32 (2012), 1901-1914. doi: 10.3934/dcds.2012.32.1901.
[44]	Y. Tao and M. Winkler, Locally bounded global solutions in a three-dimensional Chemotaxis-Stokes system with nonlinear diffusion, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 157-178. doi: 10.1016/j.anihpc.2012.07.002.
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Global solutions to Chemotaxis-Navier-Stokes equations in critical Besov spaces

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