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

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

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  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.
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:
[1]

M. ChaeK. 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. ChaeK. 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

[4]

Y. Chemin, Théorémes dunicité pour le systéme de Navier-Stokes tridimensionnal, J. Anal. Math., 77 (1999), 27-50.  doi: 10.1007/BF02791256.  Google Scholar

[5]

Y. CheminM. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

Y. Chung and K. Kang, Existence of global solutions for a Keller-Segel-fluid equations with nonlinear diffusion, arXiv: 1504.02274. Google Scholar

[9]

P. ConstantinA. KiselevL. Ryzhik and A. Zlatoš, Diffusion and mixing in fluid flow, Ann. Math., 168 (2008), 643-674.  doi: 10.4007/annals.2008.168.643.  Google Scholar

[10]

R. Danchin, Fourier Analysis Methods for PDEs, Lecture Notes, 14, November 2005. Google Scholar

[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.  Google Scholar

[12]

R. DuanA. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

M. FrancescoA. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

M. Herrero and L. Velazquez, A blow-up mechanism for chemotaxis model, Ann. Sc. Norm. Super. Pisa., 24 (1997), 633-683.   Google Scholar

[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.  Google Scholar

[20]

J. HuangM. 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.  Google Scholar

[21]

T. Kato, Strong Lp-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.  Google Scholar

[22]

E. Keller and L. Segel, Initiation of slide mold aggregation viewd as an instability, J. Theor. Biol., 6 (1970), 399-415.   Google Scholar

[23]

E. Keller and L. Segel, Model for chemotaxis, J. Theor. Biol., 30 (1971), 225-234.  doi: 10.1016/0022-5193(71)90050-6.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248.  doi: 10.1007/BF02547354.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

Q. LiuT. 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.  Google Scholar

[33]

A. Lorz, Coupled chemotaxis fuid model, Math. Models Methods Appl. Sci., 20 (2010), 987-1004.  doi: 10.1142/S0218202510004507.  Google Scholar

[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.  Google Scholar

[35]

Y. MinsukL. 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

[36]

T. NagaiT. Senba and K. Yoshida, Applications of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial Ekvac., 40 (1997), 411-433.   Google Scholar

[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

[40]

C. Patlak, Random walk with persistence and external bias, Bull. Math. Biol. Biophys., 15 (1953), 311-338.  doi: 10.1007/BF02476407.  Google Scholar

[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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[45]

Y. Tao and M. Winkler, Blow-up prevention by quadratic degradation in a two-dimensional Keller-Segel-Navier-Stokes system, Z. Angew. Math. Phys., 67 (2016), Art. 138, 23 pp. doi: 10.1007/s00033-016-0732-1.  Google Scholar

[46]

Y. Tao and M. Winkler, Eventual smoothness and stabilization of large-data solutions in a three-dimensional chemotaxis system with consumption of chemoattractant, J. Differential Equations, 252 (2012), 2520-2543.  doi: 10.1016/j.jde.2011.07.010.  Google Scholar

[47]

I. TuvalL. CisnerosC. DombrowskiC. W. WolgemuthJ. O. Kessler and R. E. Goldstein, Bacterial swimming and oxygen transport near contact lines, Proc. Natl. Acad. Sci. USA, 102 (2005), 2277-2282.  doi: 10.1073/pnas.0406724102.  Google Scholar

[48]

P. Wang and D. Zhang, Convexity of Level Sets of Minimal Graph on Space Form with Nonnegative Curvature, J. Differential Equations, 262 (2017), 5534-5564.  doi: 10.1016/j.jde.2017.02.010.  Google Scholar

[49]

M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2905.  doi: 10.1016/j.jde.2010.02.008.  Google Scholar

[50]

M. Winkler, Stabilization in a two-dimensional Chemotaxis-Navier-Stokes system, Arch. Ration. Mech. Anal., 211 (2014), 455-487.  doi: 10.1007/s00205-013-0678-9.  Google Scholar

[51]

M. Winkler, Boundedness and large time behavior in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion and general sensitivity, Comm. Partial Differ. Equ., 54 (2015), 3789-3828.  doi: 10.1007/s00526-015-0922-2.  Google Scholar

[52]

M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767.  doi: 10.1016/j.matpur.2013.01.020.  Google Scholar

[53]

M. Winkler, Global large-data solutions in a Chemotaxis-(Navier-)Stokes system modeling cellular swimming in fuid drops, Comm. Partial Differ. Equ., 37 (2012), 319-351.  doi: 10.1080/03605302.2011.591865.  Google Scholar

[54]

M. Winkler, Global weak solutions in a three-dimensional Chemotaxis-Navier-Stokes system, Annales de l'Institut Henri Poincaré (C) Non Linear Analysis, 33 (2016), 1329-1352.  doi: 10.1016/j.anihpc.2015.05.002.  Google Scholar

[55]

M. Yamazaki, Solutions in the Morrey spaces of the Navier-Stokes equation with time-dependent external force, Funkcial. Ekvac., 43 (2000), 419-460.   Google Scholar

[56]

M. Yamazaki, The Navier-Stokes equations in the weak-$L^n$ space with time-dependent external force, Math. Ann., 317 (2000), 635-675.  doi: 10.1007/PL00004418.  Google Scholar

[57]

M. Yang, Z. Fu and S. Liu, Analyticity and existence of the Keller-Segel-Navier-Stokes equations in critical Besov spaces, Adv. Nonlinear Stud., 18 (2018), 517-535. doi: 10.1515/ans-2017-6046.  Google Scholar

[58]

M. YangZ. Fu and J. Sun, Existence and Gevrey regularity for a two-species chemotaxis system in homogeneous Besov spaces, Sci. China Math., 60 (2017), 1837-1856.  doi: 10.1007/s11425-016-0490-y.  Google Scholar

[59]

M. Yang and J. Sun, Gevrey regularity and Existence of Navier-Stokes-Nernst-Planck-Poisson system in critical Besov spaces, Commun. Pure Appl. Anal., 16 (2017), 1617-1639.  doi: 10.3934/cpaa.2017078.  Google Scholar

[60]

C. Zhai and T. Zhang, Global well-posedness to the 3-D incompressible inhomogeneous Navier-Stokes equations with a class of large velocity, J. Math. Phys. , 56 (2015), 091512, 18 pp. doi: 10.1063/1.4931467.  Google Scholar

[61]

Q. Zhang, Local well-posedness for the Chemotaxis-Navier-Stokes equations in Besov spaces, Nonlinear Anal. Real World Appl., 17 (2014), 89-100.  doi: 10.1016/j.nonrwa.2013.10.008.  Google Scholar

[62]

Q. Zhang and Y. Li, Global weak solutions for the three-dimensional Chemotaxis-NavierStokes system with nonlinear diffusion, J. Differential Equations, 259 (2015), 3730-3754.  doi: 10.1016/j.jde.2015.05.012.  Google Scholar

[63]

Q. Zhang and Y. Li, Convergence rates of solutions for a two-dimensional Chemotaxis-Navier-Stokes system, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2751-2759.  doi: 10.3934/dcdsb.2015.20.2751.  Google Scholar

[64]

Q. Zhang and X. Zheng, Global well-posedness for the two-dimensional incompressible Chemotaxis-Navier-Stokes equations, SIAM J. Math. Anal., 46 (2014), 3078-3105.  doi: 10.1137/130936920.  Google Scholar

[65]

T. Zhang, Global wellposedness problem for the 3-D incompressible anisotropic Navier-Stokes equations in an anisotropic space, Comm. Math. Phys., 287 (2009), 211-224.  doi: 10.1007/s00220-008-0631-1.  Google Scholar

show all references

References:
[1]

M. ChaeK. 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. ChaeK. 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

[4]

Y. Chemin, Théorémes dunicité pour le systéme de Navier-Stokes tridimensionnal, J. Anal. Math., 77 (1999), 27-50.  doi: 10.1007/BF02791256.  Google Scholar

[5]

Y. CheminM. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

Y. Chung and K. Kang, Existence of global solutions for a Keller-Segel-fluid equations with nonlinear diffusion, arXiv: 1504.02274. Google Scholar

[9]

P. ConstantinA. KiselevL. Ryzhik and A. Zlatoš, Diffusion and mixing in fluid flow, Ann. Math., 168 (2008), 643-674.  doi: 10.4007/annals.2008.168.643.  Google Scholar

[10]

R. Danchin, Fourier Analysis Methods for PDEs, Lecture Notes, 14, November 2005. Google Scholar

[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.  Google Scholar

[12]

R. DuanA. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

M. FrancescoA. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

M. Herrero and L. Velazquez, A blow-up mechanism for chemotaxis model, Ann. Sc. Norm. Super. Pisa., 24 (1997), 633-683.   Google Scholar

[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.  Google Scholar

[20]

J. HuangM. 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.  Google Scholar

[21]

T. Kato, Strong Lp-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.  Google Scholar

[22]

E. Keller and L. Segel, Initiation of slide mold aggregation viewd as an instability, J. Theor. Biol., 6 (1970), 399-415.   Google Scholar

[23]

E. Keller and L. Segel, Model for chemotaxis, J. Theor. Biol., 30 (1971), 225-234.  doi: 10.1016/0022-5193(71)90050-6.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248.  doi: 10.1007/BF02547354.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

Q. LiuT. 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.  Google Scholar

[33]

A. Lorz, Coupled chemotaxis fuid model, Math. Models Methods Appl. Sci., 20 (2010), 987-1004.  doi: 10.1142/S0218202510004507.  Google Scholar

[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.  Google Scholar

[35]

Y. MinsukL. 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

[36]

T. NagaiT. Senba and K. Yoshida, Applications of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial Ekvac., 40 (1997), 411-433.   Google Scholar

[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

[40]

C. Patlak, Random walk with persistence and external bias, Bull. Math. Biol. Biophys., 15 (1953), 311-338.  doi: 10.1007/BF02476407.  Google Scholar

[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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[45]

Y. Tao and M. Winkler, Blow-up prevention by quadratic degradation in a two-dimensional Keller-Segel-Navier-Stokes system, Z. Angew. Math. Phys., 67 (2016), Art. 138, 23 pp. doi: 10.1007/s00033-016-0732-1.  Google Scholar

[46]

Y. Tao and M. Winkler, Eventual smoothness and stabilization of large-data solutions in a three-dimensional chemotaxis system with consumption of chemoattractant, J. Differential Equations, 252 (2012), 2520-2543.  doi: 10.1016/j.jde.2011.07.010.  Google Scholar

[47]

I. TuvalL. CisnerosC. DombrowskiC. W. WolgemuthJ. O. Kessler and R. E. Goldstein, Bacterial swimming and oxygen transport near contact lines, Proc. Natl. Acad. Sci. USA, 102 (2005), 2277-2282.  doi: 10.1073/pnas.0406724102.  Google Scholar

[48]

P. Wang and D. Zhang, Convexity of Level Sets of Minimal Graph on Space Form with Nonnegative Curvature, J. Differential Equations, 262 (2017), 5534-5564.  doi: 10.1016/j.jde.2017.02.010.  Google Scholar

[49]

M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2905.  doi: 10.1016/j.jde.2010.02.008.  Google Scholar

[50]

M. Winkler, Stabilization in a two-dimensional Chemotaxis-Navier-Stokes system, Arch. Ration. Mech. Anal., 211 (2014), 455-487.  doi: 10.1007/s00205-013-0678-9.  Google Scholar

[51]

M. Winkler, Boundedness and large time behavior in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion and general sensitivity, Comm. Partial Differ. Equ., 54 (2015), 3789-3828.  doi: 10.1007/s00526-015-0922-2.  Google Scholar

[52]

M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767.  doi: 10.1016/j.matpur.2013.01.020.  Google Scholar

[53]

M. Winkler, Global large-data solutions in a Chemotaxis-(Navier-)Stokes system modeling cellular swimming in fuid drops, Comm. Partial Differ. Equ., 37 (2012), 319-351.  doi: 10.1080/03605302.2011.591865.  Google Scholar

[54]

M. Winkler, Global weak solutions in a three-dimensional Chemotaxis-Navier-Stokes system, Annales de l'Institut Henri Poincaré (C) Non Linear Analysis, 33 (2016), 1329-1352.  doi: 10.1016/j.anihpc.2015.05.002.  Google Scholar

[55]

M. Yamazaki, Solutions in the Morrey spaces of the Navier-Stokes equation with time-dependent external force, Funkcial. Ekvac., 43 (2000), 419-460.   Google Scholar

[56]

M. Yamazaki, The Navier-Stokes equations in the weak-$L^n$ space with time-dependent external force, Math. Ann., 317 (2000), 635-675.  doi: 10.1007/PL00004418.  Google Scholar

[57]

M. Yang, Z. Fu and S. Liu, Analyticity and existence of the Keller-Segel-Navier-Stokes equations in critical Besov spaces, Adv. Nonlinear Stud., 18 (2018), 517-535. doi: 10.1515/ans-2017-6046.  Google Scholar

[58]

M. YangZ. Fu and J. Sun, Existence and Gevrey regularity for a two-species chemotaxis system in homogeneous Besov spaces, Sci. China Math., 60 (2017), 1837-1856.  doi: 10.1007/s11425-016-0490-y.  Google Scholar

[59]

M. Yang and J. Sun, Gevrey regularity and Existence of Navier-Stokes-Nernst-Planck-Poisson system in critical Besov spaces, Commun. Pure Appl. Anal., 16 (2017), 1617-1639.  doi: 10.3934/cpaa.2017078.  Google Scholar

[60]

C. Zhai and T. Zhang, Global well-posedness to the 3-D incompressible inhomogeneous Navier-Stokes equations with a class of large velocity, J. Math. Phys. , 56 (2015), 091512, 18 pp. doi: 10.1063/1.4931467.  Google Scholar

[61]

Q. Zhang, Local well-posedness for the Chemotaxis-Navier-Stokes equations in Besov spaces, Nonlinear Anal. Real World Appl., 17 (2014), 89-100.  doi: 10.1016/j.nonrwa.2013.10.008.  Google Scholar

[62]

Q. Zhang and Y. Li, Global weak solutions for the three-dimensional Chemotaxis-NavierStokes system with nonlinear diffusion, J. Differential Equations, 259 (2015), 3730-3754.  doi: 10.1016/j.jde.2015.05.012.  Google Scholar

[63]

Q. Zhang and Y. Li, Convergence rates of solutions for a two-dimensional Chemotaxis-Navier-Stokes system, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2751-2759.  doi: 10.3934/dcdsb.2015.20.2751.  Google Scholar

[64]

Q. Zhang and X. Zheng, Global well-posedness for the two-dimensional incompressible Chemotaxis-Navier-Stokes equations, SIAM J. Math. Anal., 46 (2014), 3078-3105.  doi: 10.1137/130936920.  Google Scholar

[65]

T. Zhang, Global wellposedness problem for the 3-D incompressible anisotropic Navier-Stokes equations in an anisotropic space, Comm. Math. Phys., 287 (2009), 211-224.  doi: 10.1007/s00220-008-0631-1.  Google Scholar

[1]

Laiqing Meng, Jia Yuan, Xiaoxin Zheng. Global existence of almost energy solution to the two-dimensional chemotaxis-Navier-Stokes equations with partial diffusion. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3413-3441. doi: 10.3934/dcds.2019141

[2]

Radjesvarane Alexandre, Mouhamad Elsafadi. Littlewood-Paley theory and regularity issues in Boltzmann homogeneous equations II. Non cutoff case and non Maxwellian molecules. Discrete & Continuous Dynamical Systems - A, 2009, 24 (1) : 1-11. doi: 10.3934/dcds.2009.24.1

[3]

Xiaoping Zhai, Zhaoyang Yin. Global solutions to the Chemotaxis-Navier-Stokes equations with some large initial data. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2829-2859. doi: 10.3934/dcds.2017122

[4]

Yulan Wang. Global solvability in a two-dimensional self-consistent chemotaxis-Navier-Stokes system. Discrete & Continuous Dynamical Systems - S, 2020, 13 (2) : 329-349. doi: 10.3934/dcdss.2020019

[5]

Sachiko Ishida. Global existence and boundedness for chemotaxis-Navier-Stokes systems with position-dependent sensitivity in 2D bounded domains. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3463-3482. doi: 10.3934/dcds.2015.35.3463

[6]

Qingshan Zhang, Yuxiang Li. Convergence rates of solutions for a two-dimensional chemotaxis-Navier-Stokes system. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2751-2759. doi: 10.3934/dcdsb.2015.20.2751

[7]

Hai-Yang Jin, Tian Xiang. Convergence rates of solutions for a two-species chemotaxis-Navier-Stokes sytstem with competitive kinetics. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1919-1942. doi: 10.3934/dcdsb.2018249

[8]

J. Huang, Marius Paicu. Decay estimates of global solution to 2D incompressible Navier-Stokes equations with variable viscosity. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4647-4669. doi: 10.3934/dcds.2014.34.4647

[9]

Lihuai Du, Ting Zhang. Local and global strong solution to the stochastic 3-D incompressible anisotropic Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4745-4765. doi: 10.3934/dcds.2018209

[10]

Qi S. Zhang. An example of large global smooth solution of 3 dimensional Navier-Stokes equations without pressure. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5521-5523. doi: 10.3934/dcds.2013.33.5521

[11]

Zhenhua Guo, Zilai Li. Global existence of weak solution to the free boundary problem for compressible Navier-Stokes. Kinetic & Related Models, 2016, 9 (1) : 75-103. doi: 10.3934/krm.2016.9.75

[12]

Guangwu Wang, Boling Guo. Global weak solution to the quantum Navier-Stokes-Landau-Lifshitz equations with density-dependent viscosity. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 6141-6166. doi: 10.3934/dcdsb.2019133

[13]

Hongmei Cao, Hao-Guang Li, Chao-Jiang Xu, Jiang Xu. Well-posedness of Cauchy problem for Landau equation in critical Besov space. Kinetic & Related Models, 2019, 12 (4) : 829-884. doi: 10.3934/krm.2019032

[14]

Hi Jun Choe, Bataa Lkhagvasuren, Minsuk Yang. Wellposedness of the Keller-Segel Navier-Stokes equations in the critical Besov spaces. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2453-2464. doi: 10.3934/cpaa.2015.14.2453

[15]

Daoyuan Fang, Chenyin Qian. Regularity criterion for 3D Navier-Stokes equations in Besov spaces. Communications on Pure & Applied Analysis, 2014, 13 (2) : 585-603. doi: 10.3934/cpaa.2014.13.585

[16]

Minghua Yang, Jinyi Sun. Gevrey regularity and existence of Navier-Stokes-Nernst-Planck-Poisson system in critical Besov spaces. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1617-1639. doi: 10.3934/cpaa.2017078

[17]

Jingrui Wang, Keyan Wang. Almost sure existence of global weak solutions to the 3D incompressible Navier-Stokes equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 5003-5019. doi: 10.3934/dcds.2017215

[18]

Chérif Amrouche, Nour El Houda Seloula. $L^p$-theory for the Navier-Stokes equations with pressure boundary conditions. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1113-1137. doi: 10.3934/dcdss.2013.6.1113

[19]

Chérif Amrouche, María Ángeles Rodríguez-Bellido. On the very weak solution for the Oseen and Navier-Stokes equations. Discrete & Continuous Dynamical Systems - S, 2010, 3 (2) : 159-183. doi: 10.3934/dcdss.2010.3.159

[20]

Wenjing Song, Ganshan Yang. The regularization of solution for the coupled Navier-Stokes and Maxwell equations. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 2113-2127. doi: 10.3934/dcdss.2016087

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (101)
  • HTML views (132)
  • Cited by (0)

Other articles
by authors

[Back to Top]