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February  2022, 21(2): 471-492. doi: 10.3934/cpaa.2021184

Global well-posedness in a chemotaxis system with oxygen consumption

Xujie Yang

School of Mathematical Sciences, University of Electronic, Science and Technology of China, Chengdu, 611731, China

Received  May 2021 Revised  August 2021 Published  February 2022 Early access  November 2021

Fund Project: This work is supported by the Applied Fundamental Research Program of Sichuan Province (no. 2020YJ0264)

Motivated by the studies of the hydrodynamics of the tethered bacteria Thiovulum majus in a liquid environment, we consider the following chemotaxis system
$ \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*} $
under homogeneous Neumann boundary conditions in a bounded convex domain
$ \Omega\subset \mathbb{R}^d(d\in\{2, 3\}) $
 with smooth boundary. For any given fluid
$ {\bf u} $
, it is proved that if
$ d = 2 $
, the corresponding initial-boundary value problem admits a unique global classical solution which is uniformly bounded, while if
$ d = 3 $
, such solution still exists under the additional condition that
$ 0<\chi\leq \frac{1}{16\|c(\cdot, 0)\|_{L^\infty(\Omega)}} $
.
Keywords: Chemotaxis, oxygen consumption, global boundedness, interpolation inequality, weighted function.
Mathematics Subject Classification: Primary: 35K55, 35Q92, 35Q35; Secondary: 92C17.
Citation: Xujie Yang. Global well-posedness in a chemotaxis system with oxygen consumption. Communications on Pure and Applied Analysis, 2022, 21 (2) : 471-492. doi: 10.3934/cpaa.2021184
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.

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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