No Access

Stationary solutions to a chemotaxis-consumption model with realistic boundary conditions for the oxygen

Institute for Analysis and Scientific Computing, TU Wien, Wiedner Hauptstraße 8-10, 1040 Wien, Austria

E-mail Address: marcel.braukhoff@asc.tuwien.ac.at

and

Department of Applied Mathematics and Statistics, Faculty of Mathematics, Physics and Informatics, Comenius University, Mlynská Dolina, 84248 Bratislava, Slovakia

Institut für Mathematik, Universität Paderborn, Warburger Straße 100, 33098 Paderborn, Germany

E-mail Address: jlankeit@math.uni-paderborn.de

Corresponding author

Search for more papers by this author

https://doi.org/10.1142/S0218202519500398Cited by:34 (Source: Crossref)

Abstract

Previous studies of chemotaxis models with consumption of the chemoattractant (with or without fluid) have not been successful in explaining pattern formation even in the simplest form of concentration near the boundary, which had been experimentally observed. Following the suggestions that the main reason for that is the usage of inappropriate boundary conditions, in this paper we study the solutions to the stationary chemotaxis system

{0 = Δ n - \nabla \cdot (n \nabla c), 0 = Δ c - n c <math display="block" altimg="eq-00001.gif"><mrow><mfenced separators="" open="{" close=""><mrow><mtable equalrows="false" columnlines="" equalcolumns="false" class="array"><mtr><mtd class="array" columnalign="left"><mn>0</mn><mo class="MathClass-rel">=</mo><mi mathvariant="normal">Δ</mi><mi>n</mi><mo stretchy="false" class="MathClass-bin">-</mo><mo class="MathClass-op">\nabla</mo><mo stretchy="false" class="MathClass-bin">\cdot</mo><mo class="MathClass-open" stretchy="false">(</mo><mi>n</mi><mo class="MathClass-op">\nabla</mo><mi>c</mi><mo class="MathClass-close" stretchy="false">)</mo><mo class="MathClass-punc">,</mo></mtd></mtr><mtr><mtd class="array" columnalign="left"><mn>0</mn><mo class="MathClass-rel">=</mo><mi mathvariant="normal">Δ</mi><mi>c</mi><mo stretchy="false" class="MathClass-bin">-</mo><mi>n</mi><mi>c</mi></mtd></mtr></mtable></mrow></mfenced></mrow></math>

in bounded domains

$Ω \subset ℝ^{N}$ ,

$N \geq 1$ , under the no-flux boundary conditions for

$n$ and the physically meaningful condition

\partial ν c = (γ - c) g <math display="block" altimg="eq-00005.gif"><mrow><msub><mrow><mi>\partial</mi></mrow><mrow><mi>ν</mi></mrow></msub><mi>c</mi><mo class="MathClass-rel">=</mo><mo class="MathClass-open" stretchy="false">(</mo><mi>γ</mi><mo stretchy="false" class="MathClass-bin">-</mo><mi>c</mi><mo class="MathClass-close" stretchy="false">)</mo><mi>g</mi></mrow></math>

$c$ , with the given parameter

$γ > 0$ and

$g \in C^{1 + β_{*}} (Ω)$ ,

$β_{*} \in (0, 1)$ , satisfying

$g \geq 0$ ,

$g ≢ 0$ on

$\partial Ω$ . We prove the existence and uniqueness of solutions for any given mass

$\int_{Ω} n > 0$ . These solutions are nonconstant.

Communicated by N. Bellomo

Keywords:

AMSC: 35Q92, 92C17, 35J57, 35A02

References

1. P. W. Atkins and J. de Paula , Atkins’ Physical Chemistry, 8th edn. (Oxford University Press, 2006). Google Scholar
2. N. Bellomo, A. Bellouquid, Y. Tao and M. Winkler , Toward a mathematical theory of Keller–Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci. 25 (2015) 1663–1763. Link, Web of Science, Google Scholar
3. P. Biler , Local and global solvability of some parabolic systems modelling chemotaxis, Adv. Math. Sci. Appl. 8 (1998) 715–743. Google Scholar
4. T. Black, J. Lankeit and M. Mizukami , A Keller–Segel-fluid system with singular sensitivity: Generalized solutions, Math. Methods Appl. Sci. 42 (2019) 3002–3020. Crossref, Web of Science, Google Scholar
5. M. Braukhoff , Global (weak) solution of the chemotaxis-Navier–Stokes equations with non-homogeneous boundary conditions and logistic growth, Ann. Inst. Henri Poincaré, Non Linéaire 34 (2017) 1013–1039. Crossref, Web of Science, Google Scholar
6. 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 Differ. Equ. 55 (2016) 107-1–107-39. Crossref, Web of Science, Google Scholar
7. 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. Crossref, Web of Science, Google Scholar
8. A. Chertock, K. Fellner, A. Kurganov, A. Lorz and P. A. Markowich , Sinking, merging and stationary plumes in a coupled chemotaxis-fluid model: A high-resolution numerical approach, J. Fluid Mech. 694 (2012) 155–190. Crossref, Web of Science, Google Scholar
9. M. del Pino, A. Pistoia and G. Vaira , Large mass boundary condensation patterns in the stationary Keller–Segel system, J. Differ. Equ. 261 (2016) 3414–3462. Crossref, Web of Science, Google Scholar
10. M. Di Francesco, A. Lorz and P. A. Markowich , Chemotaxis-fluid coupled model for swimming bacteria with nonlinear diffusion: Global existence and asymptotic behavior, Discrete Contin. Dyn. Syst. 28 (2010) 1437–1453. Crossref, Web of Science, Google Scholar
11. C. Dombrowski, L. Cisneros, S. Chatkaew, R. E. Goldstein and J. O. Kessler , Self-concentration and large-scale coherence in bacterial dynamics, Phys. Rev. Lett. 93 (2004) 098103. Crossref, Web of Science, Google Scholar
12. L. Fan and H.-Y. Jin , Global existence and asymptotic behavior to a chemotaxis system with consumption of chemoattractant in higher dimensions, J. Math. Phys. 58 (2017) 011503. Crossref, Web of Science, Google Scholar
13. J. Fan and K. Zhao , Global dynamics of a coupled chemotaxis-fluid model on bounded domains, J. Math. Fluid Mech. 16 (2014) 351–364. Crossref, Web of Science, Google Scholar
14. E. Feireisl, P. Laurençot and H. Petzeltová , On convergence to equilibria for the Keller–Segel chemotaxis model, J. Differ. Equ. 236 (2007) 551–569. Crossref, Web of Science, Google Scholar
15. D. Gilbarg and N. S. Trudinger , Elliptic Partial Differential Equations of Second Order, Grundlehren der Mathematischen Wissenschaften, Vol. 224 (Springer-Verlag, 1977). Crossref, Google Scholar
16. D. Horstmann , The nonsymmetric case of the Keller–Segel model in chemotaxis: Some recent results, Nonlinear Differ. Equ. Appl. NoDEA 8 (2001) 399–423. Crossref, Web of Science, Google Scholar
17. J. Jiang, H. Wu and S. Zheng , Global existence and asymptotic behavior of solutions to a chemotaxis-fluid system on general bounded domains, Asymptotic Anal. 92 (2015) 249–258. Crossref, Web of Science, Google Scholar
18. Y. Kabeya and W.-M. Ni , Stationary Keller–Segel model with the linear sensitivity (Variational problems and related topics), Sūrikaisekikenkyūsho Kōkyūroku 1025 (1998) 44–65. Google Scholar
19. P. Knosalla , Global solutions of aerotaxis equations, Appl. Math. (Warsaw) 44 (2017) 135–148. Crossref, Google Scholar
20. P. Knosalla and T. Nadzieja , Stationary solutions of aerotaxis equations, Appl. Math. (Warsaw) 42 (2015) 125–135. Crossref, Google Scholar
21. J. Lankeit , Chemotaxis can prevent thresholds on population density, Discrete Contin. Dyn. Syst. B 20 (2015) 1499–1527. Crossref, Web of Science, Google Scholar
22. J. Lankeit , Long-term behaviour in a chemotaxis-fluid system with logistic source, Math. Models Methods Appl. Sci. 26 (2016) 2071–2109. Link, Web of Science, Google Scholar
23. J. Lankeit and Y. Wang , Global existence, boundedness and stabilization in a high-dimensional chemotaxis system with consumption, Discrete Contin. Dyn. Syst. 37 (2017) 6099–6121. Crossref, Web of Science, Google Scholar
24. H. G. Lee and J. Kim , Numerical investigation of falling bacterial plumes caused by bioconvection in a three-dimensional chamber, Eur. J. Mech. B, Fluids 52 (2015) 120–130. Crossref, Web of Science, Google Scholar
25. T. Li, A. Suen, M. Winkler and C. Xue , Global small-data solutions of a two-dimensional chemotaxis system with rotational flux terms, Math. Models Methods Appl. Sci. 25 (2015) 721–746. Link, Web of Science, Google Scholar
26. C.-S. Lin, W.-M. Ni and I. Takagi , Large amplitude stationary solutions to a chemotaxis system, J. Differ. Equ. 72 (1988) 1–27. Crossref, Web of Science, Google Scholar
27. A. Lorz , Coupled chemotaxis fluid model, Math. Models Methods Appl. Sci. 20 (2010) 987–1004. Link, Web of Science, Google Scholar
28. N. S. Nadirashvili , On a problem with oblique derivative, Math. USSR-Sb. 55 (1986) 397. Crossref, Google Scholar
29. K. J. Painter and T. Hillen , Spatio-temporal chaos in a chemotaxis model, Physica D 240 (2011) 363–375. Crossref, Web of Science, Google Scholar
30. Y. Peng and Z. Xiang , Global existence and convergence rates to a chemotaxis-fluids system with mixed boundary conditions, J. Differential Equations 267, https://doi.org/10.1016/j.jde.2019.02.007. Web of Science, Google Scholar
31. R. Schaaf , Stationary solutions of chemotaxis systems, Trans. Amer. Math. Soc. 292 (1985) 531–556. Crossref, Web of Science, Google Scholar
32. J. Schöberl , NETGEN An advancing front 2D/3D-mesh generator based on abstract rules, Comput. Vis. Science 1 (1997) 41–52. Crossref, Google Scholar
33. J. Schöberl, C++11 implementation of finite elements in NGSolve, ASC Report No. 30/2014, Institute for Analysis and Scientific Computing, Vienna University of Technology (2014). Google Scholar
34. T. Senba and T. Suzuki , Some structures of the solution set for a stationary system of chemotaxis, Adv. Math. Sci. Appl. 10 (2000) 191–224. Google Scholar
35. Z. Tan and J. Zhou , Decay estimate of solutions to the coupled chemotaxis-fluid equations in $ℝ^{3}$ , Nonlinear Anal., Real World Appl. 43 (2018) 323–347. Crossref, Web of Science, Google Scholar
36. Y. Tao , Boundedness in a chemotaxis model with oxygen consumption by bacteria, J. Math. Anal. Appl. 381 (2011) 521–529. Crossref, Web of Science, Google Scholar
37. 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. Crossref, Web of Science, Google Scholar
38. I. Tuval, L. Cisneros, C. Dombrowski, C. W. Wolgemuth, J. O. Kessler and R. E. Goldstein , Bacterial swimming and oxygen transport near contact lines, Proc. Natl. Acad. Sci. USA 102 (2005) 2277–2282. Crossref, Web of Science, Google Scholar
39. G. Wang and J. Wei , Steady state solutions of a reaction-diffusion system modeling chemotaxis, Math. Nachr. 233–234 (2002) 221–236. Crossref, Web of Science, Google Scholar
40. M. Winkler , How far can chemotactic cross-diffusion enforce exceeding carrying capacities? J. Nonlinear Sci. 24 (2014) 809–855. Crossref, Web of Science, Google Scholar
41. M. Winkler , Stabilization in a two-dimensional chemotaxis-Navier–Stokes system, Arch. Ration. Mech. Anal. 211 (2014) 455–487. Crossref, Web of Science, Google Scholar
42. M. Winkler , Boundedness and large time behavior in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion and general sensitivity, Calc. Var. Partial Differ. Equ. 54 (2015) 3789–3828. Crossref, Web of Science, Google Scholar
43. M. Winkler , Asymptotic homogenization in a three-dimensional nutrient taxis system involving food-supported proliferation, J. Differ. Equ. 263 (2017) 4826–4869. Crossref, Web of Science, Google Scholar
44. M. Winkler , How far do chemotaxis-driven forces influence regularity in the Navier–Stokes system? Trans. Amer. Math. Soc. 369 (2017) 3067–3125. Crossref, Web of Science, Google Scholar
45. X. Ye , Existence and decay of global smooth solutions to the coupled chemotaxis-fluid model, J. Math. Anal. Appl. 427 (2015) 60–73. Crossref, Web of Science, Google Scholar
46. Q. Zhang and Y. Li , Convergence rates of solutions for a two-dimensional chemotaxis-Navier–Stokes system, Discrete Contin. Dyn. Syst. B 20 (2015) 2751–2759. Crossref, Web of Science, Google Scholar