Global well-posedness in a chemotaxis system with oxygen consumption

Xujie Yang

doi:10.3934/cpaa.2021184

Article Contents

2022, Volume 21, Issue 2: 471-492. Doi: 10.3934/cpaa.2021184

This issue Previous Article Variation and oscillation for harmonic operators in the inverse Gaussian setting Next Article Global well-posedness and exponential decay for 3D nonhomogeneous magneto-micropolar fluid equations with vacuum

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

Early access: November 2021

Published: February 2022

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

Abstract / Introduction Full Text(HTML) Related Papers Cited by

Abstract

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:

Mathematics Subject Classification: Primary: 35K55, 35Q92, 35Q35; Secondary: 92C17.

Citation:

Full Text(HTML)

Related Papers

Cited by

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.