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Global well-posedness in a chemotaxis system with oxygen consumption
School of Mathematical Sciences, University of Electronic, Science and Technology of China, Chengdu, 611731, China |
$ \begin{equation*} \left\{ \begin{split} & n_t = \Delta n-\nabla\cdot\left(n\chi(c)\nabla{c}\right)+nc, &x\in \Omega, t>0, \ & c_t = \Delta c-{\bf u}\cdot\nabla c-nc, &x\in \Omega, t>0\ \end{split} \right. \end{equation*} $ |
$ \Omega\subset \mathbb{R}^d(d\in\{2, 3\}) $ |
$ {\bf u} $ |
$ d = 2 $ |
$ d = 3 $ |
$ 0<\chi\leq \frac{1}{16\|c(\cdot, 0)\|_{L^\infty(\Omega)}} $ |
References:
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X. Cao and J. Lankeit, Global classical small-data solutions for a three-dimensional chemotaxis Navier-Stokes system involving matrix-valued sensitivities, Calc. Var., 55 (2016), 39 pp.
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M. Chae, K. Kang and J. Lee,
Global existence and temporal decay in Keller-Segel models coupled to fluid equations, Commun. Partial Differ. Equ., 39 (2014), 1205-1235.
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R. Duan, A. Lorz and P. Markowich,
Global solutions to the coupled chemotaxis-fluid equations, Commun. Partial Differ. Equ., 35 (2010), 1635-1673.
doi: 10.1080/03605302.2010.497199. |
[4] |
R. Duan, X. Li and Z. Xiang,
Global existence and large time behavior for a two-dimensional chemotaxis-Navier-Stokes system, J. Differ. Equ., 263 (2017), 6284-6316.
doi: 10.1016/j.jde.2017.07.015. |
[5] |
T. Fenchel,
Motility and chemosensory behaviour of the sulphur bacterium thiovulum majus, Microbiology, 140 (1994), 3109-3116.
|
[6] |
P. He, Y. Wang and L. Zhao,
A further study on a 3D chemotaxis-Stokes system with tensor-valued sensitivity, Appl. Math. Lett., 90 (2019), 23-29.
doi: 10.1016/j.aml.2018.09.019. |
[7] |
D. Horstmann and M. Winkler,
Boundedness vs. blow-up in a chemotaxis system, J. Differ. Equ., 215 (2005), 52-107.
doi: 10.1016/j.jde.2004.10.022. |
[8] |
J. Lighthill,
Flagellar hydrodynamics: The John von Neumann Lecture, SIAM Rev., 18 (1976), 161-230.
doi: 10.1137/1018040. |
[9] |
P. L. Lions,
Résolution de problèmes elliptiques quasilinéaires, Arch. Rat. Mech. Anal., 74 (1980), 335-353.
doi: 10.1007/BF00249679. |
[10] |
A. Petroff and A. Libchaber, Hydrodynamics and collective behavior of the tethered bacterium Thiovulum majus, Proc. Natl. Acad. Sci. USA., 111 (2014), E537–E545. |
[11] |
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. |
[12] |
Y. Peng and Z. Xiang,
Global existence and convergence rates to a chemotaxis-fluids system with mixed boundary conditions, J. Differ. Equ., 267 (2019), 1277-1321.
doi: 10.1016/j.jde.2019.02.007. |
[13] |
I. Tuval, L. Cisneros, C. Dombrowski and et al.,
Bacterial swimming and oxygen transport near contact lines, Proc. Natl. Acad. Sci. USA., 102 (2005), 2277-2282.
|
[14] |
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. |
[15] |
Y. Tao and M. Winkler,
Eventual smoothness and stabilization of large-data solutions in a three-dimensional chemotaxis system with consumption of chemoattractant, J. Differ. Equ., 252 (2012), 2520-2543.
doi: 10.1016/j.jde.2011.07.010. |
[16] |
Y. Tao and M. Winkler,
Energy-type estimates and global solvability in a two-dimensional chemotaxis-haptotaxis model with remodeling of non-diffusible attractant, J. Differ. Equ., 257 (2014), 784-815.
doi: 10.1016/j.jde.2014.04.014. |
[17] |
Y. Wang, F. Pang and H. Li,
Boundedness in a three-dimensional chemotaxis-Stokes system with tensor-valued sensitivity, Comput. Math. Appl., 71 (2016), 712-722.
doi: 10.1016/j.camwa.2015.12.026. |
[18] |
Y. Wang and Z. Xiang,
Global existence and boundedness in a Keller-Segel-Stokes system involving a tensor-valued sensitivity with saturation, J. Differ. Equ., 259 (2015), 7578-7609.
doi: 10.1016/j.jde.2015.08.027. |
[19] |
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. Differ. Equ., 261 (2016), 4944-4973.
doi: 10.1016/j.jde.2016.07.010. |
[20] |
Y. Wang, M. Winkler and Z. Xiang,
Global classical solutions in a two-dimensional chemotaxis-Navier-Stokes system with subcritical sensitivity, Ann. Sc. Norm. Super. Pisa. Cl. Sci., 18 (2018), 421-466.
doi: 10.1109/tps.2017.2783887. |
[21] |
Y. Wang, M. Winkler and Z. Xiang,
The small-convection limit in a two-dimensional chemotaxis-Navier-Stokes system, Math. Z., 289 (2018), 71-108.
doi: 10.1007/s00209-017-1944-6. |
[22] |
Y. Wang, M. Winkler and Z. Xiang, The fast signal diffusion limit in Keller-Segel(-fluid) systems, Calc. Var., 58 (2019), 40 pp.
doi: 10.1007/s00526-019-1656-3. |
[23] |
Y. Wang, M. Winkler and Z. Xiang,
Immediate regularization of measure-type population densities in a two-dimensional chemotaxis system with signal consumption, Sci. China Math., 64 (2021), 725-746.
doi: 10.1007/s11425-020-1708-0. |
[24] |
Y. Wang, M. Winkler and Z. Xiang,
Local energy estimates and global solvability in a three-dimensional chemotaxis-fluid system with prescribed signal on the boundary, Commun. Partial Differ. Equ., 46 (2021), 1058-1091.
doi: 10.1080/03605302.2020.1870236. |
[25] |
M. Winkler,
Global large-date solutions in a chemotaxis-(Navier-)Stokes system modeling cellular swimming in fluid drops, Commun. Partial Differ. Equ., 37 (2012), 319-351.
doi: 10.1080/03605302.2011.591865. |
[26] |
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. |
[27] |
M. Winkler,
Global weak solutions in a three-dimensional chemotaxis-Navier-Stokes system, Ann. I. H. Poincaré-AN, 33 (2016), 1329-1352.
doi: 10.1016/j.anihpc.2015.05.002. |
[28] |
M. Winkler,
Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differ. Equ., 248 (2010), 2889-2905.
doi: 10.1016/j.jde.2010.02.008. |
[29] |
M. Winkler,
Absence of collapse in a parabolic chemotaxis system with signal-dependent sensitivity, Math. Nachr., 283 (2010), 1664-1673.
doi: 10.1002/mana.200810838. |
[30] |
C. Wu and Z. Xiang,
The small-convection limit in a two-dimensional Keller-Segel-Navier-Stokes system, J. Differ. Equ., 267 (2019), 938-978.
doi: 10.1016/j.jde.2019.01.027. |
[31] |
C. Wu and Z. Xiang,
Asymptotic dynamics on a chemotaxis-Navier-Stokes system with nonlinear diffusion and inhomogeneous boundary conditions, Math. Models Methods Appl. Sci., 30 (2020), 1325-1374.
doi: 10.1142/S0218202520500244. |
show all references
References:
[1] |
X. Cao and J. Lankeit, Global classical small-data solutions for a three-dimensional chemotaxis Navier-Stokes system involving matrix-valued sensitivities, Calc. Var., 55 (2016), 39 pp.
doi: 10.1007/s00526-016-1027-2. |
[2] |
M. Chae, K. Kang and J. Lee,
Global existence and temporal decay in Keller-Segel models coupled to fluid equations, Commun. Partial Differ. Equ., 39 (2014), 1205-1235.
doi: 10.1080/03605302.2013.852224. |
[3] |
R. Duan, A. Lorz and P. Markowich,
Global solutions to the coupled chemotaxis-fluid equations, Commun. Partial Differ. Equ., 35 (2010), 1635-1673.
doi: 10.1080/03605302.2010.497199. |
[4] |
R. Duan, X. Li and Z. Xiang,
Global existence and large time behavior for a two-dimensional chemotaxis-Navier-Stokes system, J. Differ. Equ., 263 (2017), 6284-6316.
doi: 10.1016/j.jde.2017.07.015. |
[5] |
T. Fenchel,
Motility and chemosensory behaviour of the sulphur bacterium thiovulum majus, Microbiology, 140 (1994), 3109-3116.
|
[6] |
P. He, Y. Wang and L. Zhao,
A further study on a 3D chemotaxis-Stokes system with tensor-valued sensitivity, Appl. Math. Lett., 90 (2019), 23-29.
doi: 10.1016/j.aml.2018.09.019. |
[7] |
D. Horstmann and M. Winkler,
Boundedness vs. blow-up in a chemotaxis system, J. Differ. Equ., 215 (2005), 52-107.
doi: 10.1016/j.jde.2004.10.022. |
[8] |
J. Lighthill,
Flagellar hydrodynamics: The John von Neumann Lecture, SIAM Rev., 18 (1976), 161-230.
doi: 10.1137/1018040. |
[9] |
P. L. Lions,
Résolution de problèmes elliptiques quasilinéaires, Arch. Rat. Mech. Anal., 74 (1980), 335-353.
doi: 10.1007/BF00249679. |
[10] |
A. Petroff and A. Libchaber, Hydrodynamics and collective behavior of the tethered bacterium Thiovulum majus, Proc. Natl. Acad. Sci. USA., 111 (2014), E537–E545. |
[11] |
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. |
[12] |
Y. Peng and Z. Xiang,
Global existence and convergence rates to a chemotaxis-fluids system with mixed boundary conditions, J. Differ. Equ., 267 (2019), 1277-1321.
doi: 10.1016/j.jde.2019.02.007. |
[13] |
I. Tuval, L. Cisneros, C. Dombrowski and et al.,
Bacterial swimming and oxygen transport near contact lines, Proc. Natl. Acad. Sci. USA., 102 (2005), 2277-2282.
|
[14] |
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. |
[15] |
Y. Tao and M. Winkler,
Eventual smoothness and stabilization of large-data solutions in a three-dimensional chemotaxis system with consumption of chemoattractant, J. Differ. Equ., 252 (2012), 2520-2543.
doi: 10.1016/j.jde.2011.07.010. |
[16] |
Y. Tao and M. Winkler,
Energy-type estimates and global solvability in a two-dimensional chemotaxis-haptotaxis model with remodeling of non-diffusible attractant, J. Differ. Equ., 257 (2014), 784-815.
doi: 10.1016/j.jde.2014.04.014. |
[17] |
Y. Wang, F. Pang and H. Li,
Boundedness in a three-dimensional chemotaxis-Stokes system with tensor-valued sensitivity, Comput. Math. Appl., 71 (2016), 712-722.
doi: 10.1016/j.camwa.2015.12.026. |
[18] |
Y. Wang and Z. Xiang,
Global existence and boundedness in a Keller-Segel-Stokes system involving a tensor-valued sensitivity with saturation, J. Differ. Equ., 259 (2015), 7578-7609.
doi: 10.1016/j.jde.2015.08.027. |
[19] |
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. Differ. Equ., 261 (2016), 4944-4973.
doi: 10.1016/j.jde.2016.07.010. |
[20] |
Y. Wang, M. Winkler and Z. Xiang,
Global classical solutions in a two-dimensional chemotaxis-Navier-Stokes system with subcritical sensitivity, Ann. Sc. Norm. Super. Pisa. Cl. Sci., 18 (2018), 421-466.
doi: 10.1109/tps.2017.2783887. |
[21] |
Y. Wang, M. Winkler and Z. Xiang,
The small-convection limit in a two-dimensional chemotaxis-Navier-Stokes system, Math. Z., 289 (2018), 71-108.
doi: 10.1007/s00209-017-1944-6. |
[22] |
Y. Wang, M. Winkler and Z. Xiang, The fast signal diffusion limit in Keller-Segel(-fluid) systems, Calc. Var., 58 (2019), 40 pp.
doi: 10.1007/s00526-019-1656-3. |
[23] |
Y. Wang, M. Winkler and Z. Xiang,
Immediate regularization of measure-type population densities in a two-dimensional chemotaxis system with signal consumption, Sci. China Math., 64 (2021), 725-746.
doi: 10.1007/s11425-020-1708-0. |
[24] |
Y. Wang, M. Winkler and Z. Xiang,
Local energy estimates and global solvability in a three-dimensional chemotaxis-fluid system with prescribed signal on the boundary, Commun. Partial Differ. Equ., 46 (2021), 1058-1091.
doi: 10.1080/03605302.2020.1870236. |
[25] |
M. Winkler,
Global large-date solutions in a chemotaxis-(Navier-)Stokes system modeling cellular swimming in fluid drops, Commun. Partial Differ. Equ., 37 (2012), 319-351.
doi: 10.1080/03605302.2011.591865. |
[26] |
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. |
[27] |
M. Winkler,
Global weak solutions in a three-dimensional chemotaxis-Navier-Stokes system, Ann. I. H. Poincaré-AN, 33 (2016), 1329-1352.
doi: 10.1016/j.anihpc.2015.05.002. |
[28] |
M. Winkler,
Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differ. Equ., 248 (2010), 2889-2905.
doi: 10.1016/j.jde.2010.02.008. |
[29] |
M. Winkler,
Absence of collapse in a parabolic chemotaxis system with signal-dependent sensitivity, Math. Nachr., 283 (2010), 1664-1673.
doi: 10.1002/mana.200810838. |
[30] |
C. Wu and Z. Xiang,
The small-convection limit in a two-dimensional Keller-Segel-Navier-Stokes system, J. Differ. Equ., 267 (2019), 938-978.
doi: 10.1016/j.jde.2019.01.027. |
[31] |
C. Wu and Z. Xiang,
Asymptotic dynamics on a chemotaxis-Navier-Stokes system with nonlinear diffusion and inhomogeneous boundary conditions, Math. Models Methods Appl. Sci., 30 (2020), 1325-1374.
doi: 10.1142/S0218202520500244. |
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