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On a parabolic-elliptic chemotaxis-growth system with nonlinear diffusion
Global solvability in a two-dimensional self-consistent chemotaxis-Navier-Stokes system
School of Science, Xihua University, Chengdu 610039, China |
$\left\{ \begin{array}{l}{n_t} + u \cdot \nabla n = \Delta {n^m} - \nabla \cdot (n\nabla c) + \nabla \cdot (n\nabla \phi ), \;\;\;\;\;\;(x, t) \in \Omega \times (0, T), \\{c_t} + u \cdot \nabla c = \Delta c - nc, \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(x, t) \in \Omega \times (0, T), \\{u_t} + (u \cdot \nabla )u + \nabla P = \Delta u - n\nabla \phi + n\nabla c, \;\;\;\;\;\;\;\;\;\;\;\;\;(x, t) \in \Omega \times (0, T), \\\nabla \cdot u = 0, \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(x, t) \in \Omega \times (0, T), \end{array} \right.$ |
$Ω\subset \mathbb{R}^2$ |
$m>1$ |
$m∈(\frac{3}{2}, 2]$ |
References:
[1] |
N. Bellomo, A. Bellouquid, Y. Tao and M. Winkler,
Toward a mathematical theory of KellerSegel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 1663-1763.
doi: 10.1142/S021820251550044X. |
[2] |
V. Calvez and J. A. Carrillo,
Volume effects in the Keller-Segel model: Energy estimates
preventing blow-up, J. Math.Pures Appl., 86 (2006), 155-175.
doi: 10.1016/j.matpur.2006.04.002. |
[3] |
X. Cao,
Global classical solutions in chemotaxis(-Navier)-Stokes system with rotational flux
term, J.Differential Equations, 261 (2016), 6883-6914.
doi: 10.1016/j.jde.2016.09.007. |
[4] |
X. Cao and J. Lankeit, Global classical small-data solutions for a three-dimensional chemotaxis Navier-Stokes system involving matrix-valued sensitivities,
Calc. Var. Partial Differential Equations, 55 (2016), Art. 107, 39 pp.
doi: 10.1007/s00526-016-1027-2. |
[5] |
M. Chae, K. Kang and J. Lee,
Existence of smooth solutions to coupled chemotaxis-fluid
equations, Discr. Cont. Dyn. Syst. A, 33 (2013), 2271-2297.
doi: 10.3934/dcds.2013.33.2271. |
[6] |
M. Chae, K. Kang and J. Lee,
Global Existence and temporal decay in Keller-Segel models
coupled to fluid equations, Comm. Part. Diff. Eqs., 39 (2014), 1205-1235.
doi: 10.1080/03605302.2013.852224. |
[7] |
M. Difrancesco, A. Lorz and P. A. Markowich,
Chemotaxis-fluid coupled model for swimming
bacteria with nonlinear diffusion: Global existence and asymptotic behavior, Discr. Cont. Dyn. Syst. A, 28 (2010), 1437-1453.
doi: 10.3934/dcds.2010.28.1437. |
[8] |
R. Duan, X. Li and Z. Xiang,
Global existence and large time behavior for a two dimensional
chemotaxis–Navier–Stokes system, J. Differential Equations, 263 (2017), 6284-6316.
doi: 10.1016/j.jde.2017.07.015. |
[9] |
R. Duan, A. Lorz and P. A. Markowich,
Global solutions to the coupled chemotaxis-fluid
equations, Comm. Part. Diff. Eqs., 35 (2010), 1635-1673.
doi: 10.1080/03605302.2010.497199. |
[10] |
R. Duan and Z. Xiang,
A note on global existence for the chemotaxis-Stokes model with
nonlinear diffusion, Int. Math. Res. Not. IMRN, 2014 (2014), 1833-1852.
doi: 10.1093/imrn/rns270. |
[11] |
S. Ishida,
Global existence and Boundedness for chemotaxis-Navier-Stokes system with
position-dependent sensitivity in 2D bounded domains, Discr. Cont. Dyn. Syst. A, 35 (2015), 3463-3482.
doi: 10.3934/dcds.2015.35.3463. |
[12] |
R. Kowalczyk,
Preventing blow-up in a chemotaxis model, J. Math. Anal. Appl., 305 (2005), 566-588.
doi: 10.1016/j.jmaa.2004.12.009. |
[13] |
J. Lankeit,
Long-term behaviour in a chemotaxis-fluid system with logistic source, Math. Models Methods Appl. Sci., 26 (2016), 2071-2109.
doi: 10.1142/S021820251640008X. |
[14] |
X. Li, Y. Wang and Z. Xiang,
Global existence and boundedness in a 2D Keller-Segel-Stokes
system with nonlinear diffusion and rotational flux, Commun. Math. Sci., 14 (2016), 1889-1910.
doi: 10.4310/CMS.2016.v14.n7.a5. |
[15] |
X. Li and Y. Xiao,
Global existence and boundedness in a 2D Keller-Segel-Stokes system, Nonlinear Analysis-RWA, 37 (2017), 14-30.
doi: 10.1016/j.nonrwa.2017.02.005. |
[16] |
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. |
[17] |
A. Lorz,
Coupled chemotaxis fluid equations, Math. Models Methods Appl. Sci., 20 (2010), 987-1004.
doi: 10.1142/S0218202510004507. |
[18] |
Y. Peng and Z. Xiang,
Global existence and boundedness in a 3D Keller-Segel-Stokes system with nonlinear diffusion and rotational flux,
Z. Angew. Math. Phys., 68 (2017), Art. 68, 26 pp.
doi: 10.1007/s00033-017-0816-6. |
[19] |
Y. Peng and Z. Xiang,
Global solutions to the coupled chemotaxis-fluids system in a 3D
unbounded domain with boundary, Math. Models Methods Appl. Sci., 28 (2018), 869-920.
doi: 10.1142/S0218202518500239. |
[20] |
Y. Shibata and S. Shimizu,
On the Lp − Lq maximal regularity of Neumann problem for the
Stokes equations in a bounded domain, J.Reine Angew.Math., 615 (2008), 157-209.
doi: 10.1515/CRELLE.2008.013. |
[21] |
H. Sohr,
The Navier-Stokes Equations: An Elementary Functional Analytic Approach, Birkh$\check{a}$user, Basel, 2001. |
[22] |
Z. Tan and X. Zhang,
Decay estimates of the coupled chemotaxis-fluid equations in $R^3$, J. Math. Anal. Appl., 410 (2014), 27-38.
doi: 10.1016/j.jmaa.2013.08.008. |
[23] |
Y. Tao and M. Winkler,
Global existence and boundedness in a Keller-Segel-Stokes model
with arbitrary porous medium diffusion, Discr. Cont. Dyn. Syst. A, 32 (2012), 1901-1914.
doi: 10.3934/dcds.2012.32.1901. |
[24] |
Y. Tao and M. Winkler,
Locally bounded global solutions in a three-dimensional chemotaxisStokes system with nonlinear diffusion, Ann. I. H. Poincar$\acute{e}$-AN, 30 (2013), 157-178.
doi: 10.1016/j.anihpc.2012.07.002. |
[25] |
I. Tuval, L. Cisneros and C. Dombrowski, et al., Bacterial swimming and oxygen transport
near contact lines, Proc. Nat. Acad. Sci. USA., 102 (2005), 2277–2282.
doi: 10.1073/pnas.0406724102. |
[26] |
J. L. Vázquez,
The Porous Medium Equation. Mathematical Theory, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2007. |
[27] |
D. Vorotnikov,
Weak solutions for a bioconvection model related to Bacillus subtilis, Commun. Math. Sci., 12 (2014), 545-563.
doi: 10.4310/CMS.2014.v12.n3.a8. |
[28] |
Y. Wang and X. Cao,
Global classical solutions of a 3D chemotaxis-Stokes system with rotation, Discr. Cont. Dyn. Syst. B, 20 (2015), 3235-3254.
doi: 10.3934/dcdsb.2015.20.3235. |
[29] |
Y. Wang and Z. Xiang,
Global existence and boundedness in a Keller-Segel-Stokes system
involving a tensor-valued sensitivity with saturation: The 3D case, J.Differential Equations, 261 (2016), 4944-4973.
doi: 10.1016/j.jde.2016.07.010. |
[30] |
Y. Wang,
Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system
with subcritical sensitivity, Math. Models Methods Appl. Sci., 27 (2017), 2745-2780.
doi: 10.1142/S0218202517500579. |
[31] |
Y. Wang, M. Winkler and Z. Xiang,
Global classical solutions in a two-dimensional chemotaxis-Navier-Stokes system with subcritical sensitivity, Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, 18 (2018), 421-466.
doi: 10.2422/2036-2145.201603_004. |
[32] |
Y. Wang, M. Winkler and Z. Xiang,
The small-convection limit in a two-dimensional
chemotaxis-Navier-Stokes system, Math. Zeit., 289 (2018), 71-108.
doi: 10.1007/s00209-017-1944-6. |
[33] |
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. |
[34] |
M. Winkler,
Global large-data solutions in a chemotaxis-(Navier-)Stokes system modeling
cellular swimming in fluid drops, Comm. Part. Diff. Eqs., 37 (2012), 319-351.
doi: 10.1080/03605302.2011.591865. |
[35] |
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. |
[36] |
M. Winkler,
Global mass-preserving solutions in a two-dimensional chemotaxis-Stokes system
with rotational flux components, J.Evol.Equ., 18 (2018), 1267-1289.
doi: 10.1007/s00028-018-0440-8. |
[37] |
M. Winkler,
Boundedness and large time behavior in a three-dimensional chemotaxis-Stokes
system with nonlinear diffusion and general sensitivity, Calc. Var.Partial Differential Equations, 54 (2015), 3789-3828.
doi: 10.1007/s00526-015-0922-2. |
[38] |
M. Winkler,
Global weak solutions in a three-dimensional chemotaxis-Navier-Stokes system, Ann.Inst.Henri Poincaré, Anal.Non Linéaire, 33 (2016), 1329-1352.
doi: 10.1016/j.anihpc.2015.05.002. |
[39] |
M. Winkler,
How far do chemotaxis-driven forces influence regularity in the Navier-Stokes system?, Trans.Amer.Math.Soc., 369 (2017), 3067-3125.
doi: 10.1090/tran/6733. |
[40] |
C. Xue and H. G. Othmer,
Multiscale models of taxis-driven patterning in bacterial population, SIAM J. Appl. Math., 70 (2009), 133-167.
doi: 10.1137/070711505. |
[41] |
Q. Zhang and Y. Li,
Convergence rates of solutions for a two-dimensional chemotaxis-NavierStokes system, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2751-2759.
doi: 10.3934/dcdsb.2015.20.2751. |
[42] |
Q. Zhang and Y. Li,
Global weak solutions for the three-dimensional chemotaxis-Navier-Stokes
system with nonlinear diffusion, J. Differential Equations, 259 (2015), 3730-3754.
doi: 10.1016/j.jde.2015.05.012. |
[43] |
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. |
show all references
References:
[1] |
N. Bellomo, A. Bellouquid, Y. Tao and M. Winkler,
Toward a mathematical theory of KellerSegel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 1663-1763.
doi: 10.1142/S021820251550044X. |
[2] |
V. Calvez and J. A. Carrillo,
Volume effects in the Keller-Segel model: Energy estimates
preventing blow-up, J. Math.Pures Appl., 86 (2006), 155-175.
doi: 10.1016/j.matpur.2006.04.002. |
[3] |
X. Cao,
Global classical solutions in chemotaxis(-Navier)-Stokes system with rotational flux
term, J.Differential Equations, 261 (2016), 6883-6914.
doi: 10.1016/j.jde.2016.09.007. |
[4] |
X. Cao and J. Lankeit, Global classical small-data solutions for a three-dimensional chemotaxis Navier-Stokes system involving matrix-valued sensitivities,
Calc. Var. Partial Differential Equations, 55 (2016), Art. 107, 39 pp.
doi: 10.1007/s00526-016-1027-2. |
[5] |
M. Chae, K. Kang and J. Lee,
Existence of smooth solutions to coupled chemotaxis-fluid
equations, Discr. Cont. Dyn. Syst. A, 33 (2013), 2271-2297.
doi: 10.3934/dcds.2013.33.2271. |
[6] |
M. Chae, K. Kang and J. Lee,
Global Existence and temporal decay in Keller-Segel models
coupled to fluid equations, Comm. Part. Diff. Eqs., 39 (2014), 1205-1235.
doi: 10.1080/03605302.2013.852224. |
[7] |
M. Difrancesco, A. Lorz and P. A. Markowich,
Chemotaxis-fluid coupled model for swimming
bacteria with nonlinear diffusion: Global existence and asymptotic behavior, Discr. Cont. Dyn. Syst. A, 28 (2010), 1437-1453.
doi: 10.3934/dcds.2010.28.1437. |
[8] |
R. Duan, X. Li and Z. Xiang,
Global existence and large time behavior for a two dimensional
chemotaxis–Navier–Stokes system, J. Differential Equations, 263 (2017), 6284-6316.
doi: 10.1016/j.jde.2017.07.015. |
[9] |
R. Duan, A. Lorz and P. A. Markowich,
Global solutions to the coupled chemotaxis-fluid
equations, Comm. Part. Diff. Eqs., 35 (2010), 1635-1673.
doi: 10.1080/03605302.2010.497199. |
[10] |
R. Duan and Z. Xiang,
A note on global existence for the chemotaxis-Stokes model with
nonlinear diffusion, Int. Math. Res. Not. IMRN, 2014 (2014), 1833-1852.
doi: 10.1093/imrn/rns270. |
[11] |
S. Ishida,
Global existence and Boundedness for chemotaxis-Navier-Stokes system with
position-dependent sensitivity in 2D bounded domains, Discr. Cont. Dyn. Syst. A, 35 (2015), 3463-3482.
doi: 10.3934/dcds.2015.35.3463. |
[12] |
R. Kowalczyk,
Preventing blow-up in a chemotaxis model, J. Math. Anal. Appl., 305 (2005), 566-588.
doi: 10.1016/j.jmaa.2004.12.009. |
[13] |
J. Lankeit,
Long-term behaviour in a chemotaxis-fluid system with logistic source, Math. Models Methods Appl. Sci., 26 (2016), 2071-2109.
doi: 10.1142/S021820251640008X. |
[14] |
X. Li, Y. Wang and Z. Xiang,
Global existence and boundedness in a 2D Keller-Segel-Stokes
system with nonlinear diffusion and rotational flux, Commun. Math. Sci., 14 (2016), 1889-1910.
doi: 10.4310/CMS.2016.v14.n7.a5. |
[15] |
X. Li and Y. Xiao,
Global existence and boundedness in a 2D Keller-Segel-Stokes system, Nonlinear Analysis-RWA, 37 (2017), 14-30.
doi: 10.1016/j.nonrwa.2017.02.005. |
[16] |
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. |
[17] |
A. Lorz,
Coupled chemotaxis fluid equations, Math. Models Methods Appl. Sci., 20 (2010), 987-1004.
doi: 10.1142/S0218202510004507. |
[18] |
Y. Peng and Z. Xiang,
Global existence and boundedness in a 3D Keller-Segel-Stokes system with nonlinear diffusion and rotational flux,
Z. Angew. Math. Phys., 68 (2017), Art. 68, 26 pp.
doi: 10.1007/s00033-017-0816-6. |
[19] |
Y. Peng and Z. Xiang,
Global solutions to the coupled chemotaxis-fluids system in a 3D
unbounded domain with boundary, Math. Models Methods Appl. Sci., 28 (2018), 869-920.
doi: 10.1142/S0218202518500239. |
[20] |
Y. Shibata and S. Shimizu,
On the Lp − Lq maximal regularity of Neumann problem for the
Stokes equations in a bounded domain, J.Reine Angew.Math., 615 (2008), 157-209.
doi: 10.1515/CRELLE.2008.013. |
[21] |
H. Sohr,
The Navier-Stokes Equations: An Elementary Functional Analytic Approach, Birkh$\check{a}$user, Basel, 2001. |
[22] |
Z. Tan and X. Zhang,
Decay estimates of the coupled chemotaxis-fluid equations in $R^3$, J. Math. Anal. Appl., 410 (2014), 27-38.
doi: 10.1016/j.jmaa.2013.08.008. |
[23] |
Y. Tao and M. Winkler,
Global existence and boundedness in a Keller-Segel-Stokes model
with arbitrary porous medium diffusion, Discr. Cont. Dyn. Syst. A, 32 (2012), 1901-1914.
doi: 10.3934/dcds.2012.32.1901. |
[24] |
Y. Tao and M. Winkler,
Locally bounded global solutions in a three-dimensional chemotaxisStokes system with nonlinear diffusion, Ann. I. H. Poincar$\acute{e}$-AN, 30 (2013), 157-178.
doi: 10.1016/j.anihpc.2012.07.002. |
[25] |
I. Tuval, L. Cisneros and C. Dombrowski, et al., Bacterial swimming and oxygen transport
near contact lines, Proc. Nat. Acad. Sci. USA., 102 (2005), 2277–2282.
doi: 10.1073/pnas.0406724102. |
[26] |
J. L. Vázquez,
The Porous Medium Equation. Mathematical Theory, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2007. |
[27] |
D. Vorotnikov,
Weak solutions for a bioconvection model related to Bacillus subtilis, Commun. Math. Sci., 12 (2014), 545-563.
doi: 10.4310/CMS.2014.v12.n3.a8. |
[28] |
Y. Wang and X. Cao,
Global classical solutions of a 3D chemotaxis-Stokes system with rotation, Discr. Cont. Dyn. Syst. B, 20 (2015), 3235-3254.
doi: 10.3934/dcdsb.2015.20.3235. |
[29] |
Y. Wang and Z. Xiang,
Global existence and boundedness in a Keller-Segel-Stokes system
involving a tensor-valued sensitivity with saturation: The 3D case, J.Differential Equations, 261 (2016), 4944-4973.
doi: 10.1016/j.jde.2016.07.010. |
[30] |
Y. Wang,
Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system
with subcritical sensitivity, Math. Models Methods Appl. Sci., 27 (2017), 2745-2780.
doi: 10.1142/S0218202517500579. |
[31] |
Y. Wang, M. Winkler and Z. Xiang,
Global classical solutions in a two-dimensional chemotaxis-Navier-Stokes system with subcritical sensitivity, Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, 18 (2018), 421-466.
doi: 10.2422/2036-2145.201603_004. |
[32] |
Y. Wang, M. Winkler and Z. Xiang,
The small-convection limit in a two-dimensional
chemotaxis-Navier-Stokes system, Math. Zeit., 289 (2018), 71-108.
doi: 10.1007/s00209-017-1944-6. |
[33] |
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. |
[34] |
M. Winkler,
Global large-data solutions in a chemotaxis-(Navier-)Stokes system modeling
cellular swimming in fluid drops, Comm. Part. Diff. Eqs., 37 (2012), 319-351.
doi: 10.1080/03605302.2011.591865. |
[35] |
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. |
[36] |
M. Winkler,
Global mass-preserving solutions in a two-dimensional chemotaxis-Stokes system
with rotational flux components, J.Evol.Equ., 18 (2018), 1267-1289.
doi: 10.1007/s00028-018-0440-8. |
[37] |
M. Winkler,
Boundedness and large time behavior in a three-dimensional chemotaxis-Stokes
system with nonlinear diffusion and general sensitivity, Calc. Var.Partial Differential Equations, 54 (2015), 3789-3828.
doi: 10.1007/s00526-015-0922-2. |
[38] |
M. Winkler,
Global weak solutions in a three-dimensional chemotaxis-Navier-Stokes system, Ann.Inst.Henri Poincaré, Anal.Non Linéaire, 33 (2016), 1329-1352.
doi: 10.1016/j.anihpc.2015.05.002. |
[39] |
M. Winkler,
How far do chemotaxis-driven forces influence regularity in the Navier-Stokes system?, Trans.Amer.Math.Soc., 369 (2017), 3067-3125.
doi: 10.1090/tran/6733. |
[40] |
C. Xue and H. G. Othmer,
Multiscale models of taxis-driven patterning in bacterial population, SIAM J. Appl. Math., 70 (2009), 133-167.
doi: 10.1137/070711505. |
[41] |
Q. Zhang and Y. Li,
Convergence rates of solutions for a two-dimensional chemotaxis-NavierStokes system, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2751-2759.
doi: 10.3934/dcdsb.2015.20.2751. |
[42] |
Q. Zhang and Y. Li,
Global weak solutions for the three-dimensional chemotaxis-Navier-Stokes
system with nonlinear diffusion, J. Differential Equations, 259 (2015), 3730-3754.
doi: 10.1016/j.jde.2015.05.012. |
[43] |
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. |
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