No Access

Global classical solvability and stabilization in a two-dimensional chemotaxis–fluid system with sub-logarithmic sensitivity

Ji Liu

Ji Liu

College of Sciences, Nanjing Agricultural University, No. 6 Tongwei Road, Xuanwu District, 210095 Nanjing, P. R. China

E-mail Address: liuji@njau.edu.cn

Search for more papers by this author

https://doi.org/10.1142/S0218202523400031Cited by:3 (Source: Crossref)

This article is part of the issue:

Special Issue on “Analysis of Complex Chemotaxis Models”
Guest Editors: Michael Winkler and Youshan Tao

Abstract

In this paper, we consider the following system:

in a smoothly bounded domain

$Ω \subset ℝ^{2},$ with

$κ \in {0, 1}$ and a given function

χ(c)=1c𝜃<math display="block" altimg="eq-00004.gif"><mrow><mi>χ</mi><mo class="MathClass-open" stretchy="false">(</mo><mi>c</mi><mo class="MathClass-close" stretchy="false">)</mo><mo class="MathClass-rel">=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msup><mrow><mi>c</mi></mrow><mrow><mi>𝜃</mi></mrow></msup></mrow></mfrac></mrow></math>

with

$𝜃 \in [0, 1) .$ It is proved that if

$κ = 1$ then for appropriately small initial data an associated no-flux/no-flux/Dirichlet initial-boundary value problem is globally solvable in the classical sense, and that if

$κ = 0$ then under a different but still suitable smallness restriction of the initial data, a corresponding initial-boundary value problem subject to no-flux/no-flux/Dirichlet boundary conditions admits a unique classical solution which is globally bounded and approaches a constant equilibria

$({\bar{n}}_{0}, 0, 0)$ in

$L^{\infty} (Ω) \times W^{1, \infty} (Ω) \times L^{\infty} (Ω)$ as

$t \to \infty,$ with

${\bar{n}}_{0} : = \frac{1}{| Ω |} \int_{Ω} n_{0} .$

Communicated by Michael Winkler & Youshan Tao

Keywords:

AMSC: 35B65, 35B40, 35K55, 92C17, 35Q35, 35Q92

References

1. J. Ahn, K. Kang and C. Yoon , Global classical solutions for chemotaxis–fluid systems in two dimensions, Math. Methods Appl. Sci. 44 (2021) 2254–2264. Crossref, Web of Science, Google Scholar
2. P. Biler , Global solutions to some parabolic–elliptic systems of chemotaxis, Adv. Math. Sci. Appl. 9 (1999) 347–359. Google Scholar
3. T. Black , Sublinear signal production in a two-dimensional Keller–Segel–Stokes system, Nonlinear Anal. Real World Appl. 31 (2016) 593–609. Crossref, Web of Science, Google Scholar
4. T. Black , Eventual smoothness of generalized solutions to a singular chemotaxis-Stokes system in 2D, J. Differential Equations 265 (2018) 2296–2339. Crossref, Web of Science, Google Scholar
5. E. Espejo and M. Winkler , Global classical solvability and stabilization in a two-dimensional chemotaxis-Navier–Stokes system modeling coral fertilization, Nonlinearity 31 (2018) 1227–1259. Crossref, Web of Science, Google Scholar
6. A. Friedman , Partial Differential Equations (Holt, Rinehart & Winston, 1969). Google Scholar
7. K. Fujie, A. Ito, M. Winkler and T. Yokota , Stabilization in a chemotaxis model for tumor invasion, Discrete Contin. Dyn. Syst. 36 (2016) 151–169. Web of Science, Google Scholar
8. Y. Giga , The Stokes operator in $L_{r}$ spaces, Proc. Japan Acad. Ser. A Math. Sci. 2 (1981) 85–89. Google Scholar
9. Y. Giga , Solutions for semilinear parabolic equations in $L_{p}$ and regularity of weak solutions of the Navier–Stokes system, J. Differential Equations 61 (1986) 186–212. Crossref, Web of Science, Google Scholar
10. Y. Giga and H. Sohr , Abstract $L^{p}$ estimates for the Cauchy problem with applications to the Navier–Stokes equations in exterior domains, J. Funct. Anal. 102 (1991) 72–94. Crossref, Web of Science, Google Scholar
11. D. Henry , Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, Vol. 840 (Springer-Verlag, 1981). Crossref, Google Scholar
12. Z. Jia, Z. Yang and Q. Li , Global existence, boundedness and asymptotic behavior to a chemotaxis model with singular sensitivity and logistic source, Appl. Anal. 100 (2021) 1471–1486. Crossref, Web of Science, Google Scholar
13. E. F. Keller and L. A. Segel , Traveling bands of chemotactic bacteria: A theoretical analysis, J. Theor. Biol. 30 (1971) 235–248. Crossref, Web of Science, Google Scholar
14. J. Lankeit , A new approach toward boundedness in a two-dimensional parabolic chemotaxis system with singular sensitivity, Math. Methods Appl. Sci. 39 (2016) 394–404. Crossref, Web of Science, Google Scholar
15. 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
16. J. Lankeit , Locally bounded global solutions to a chemotaxis consumption model with singular sensitivity and nonlinear diffusion, J. Differential Equations 262 (2017) 4052–4084. Crossref, Web of Science, Google Scholar
17. J. Lankeit and G. Viglialoro , Global existence and boundedness of solutions to a chemotaxis consumption model with singular sensitivity, Acta Appl. Math. 167 (2020) 75–97. Crossref, Web of Science, Google Scholar
18. H. Li and K. Zhao , Initial-boundary value problems for a system of hyperbolic balance laws arising from chemotaxis, J. Differential Equations 258 (2015) 302–308. Crossref, Web of Science, Google Scholar
19. D. Liu , Global classical solution to a chemotaxis consumption model with singular sensitivity, Nonlinear Anal. Real World Appl. 41 (2018) 497–508. Crossref, Web of Science, Google Scholar
20. J. Liu , Large-time behavior in a two-dimensional logarithmic chemotaxis-Navier–Stokes system with signal absorption, J. Evol. Equ. 21 (2021) 5135–5170. Crossref, Web of Science, Google Scholar
21. T. Nagai, T. Senba and K. Yoshida , Global existence of solutions to the parabolic systems of chemotaxis, Sūrikaisekikenkyūsho Kōkyūroku 1009 (1997) 22–28. Google Scholar
22. P. Y. H. Pang, Y. Wang and J. Yin , Asymptotic profile of a two-dimensional chemotaxis-Navier–Stokes system with singular sensitivity and logistic source, Math. Models Methods Appl. Sci. 31 (2021) 577–618. Link, Web of Science, Google Scholar
23. M. M. Porzio and V. Vespri , Holder estimates for local solutions of some doubly nonlinear degenerate parabolic equations, J. Differential Equations 103 (1993) 146–178. Crossref, Web of Science, Google Scholar
24. H. Sohr , The Navier–Stokes Equations: An Elementary Functional Analytic Approach (Birkhäuser, 2001). Crossref, Google Scholar
25. Y. Tao, L. Wang and Z. A. Wang , Large-time behavior of a parabolic–parabolic chemotaxis model with logarithmic sensitivity in one dimension, Discrete Contin. Dyn. Syst. Ser. B 257 (2013) 821–845. Google Scholar
26. 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. Differential Equations 18 (2014) 784–815. Crossref, Web of Science, Google Scholar
27. G. Viglialoro , Global existence in a two-dimensional chemotaxis-consumption model with weakly singular sensitivity, Appl. Math. Lett. 91 (2019) 121–127. Crossref, Web of Science, Google Scholar
28. Y. Wang and Z. Xiang , Global existence and boundedness in a Keller–Segel–Stokes system involving a tensor-valued sensitivity with saturation, J. Differential Equations 259 (2015) 7578–7609. Crossref, Web of Science, Google Scholar
29. Z. A. Wang, Z. Xiang and P. Yu , Asymptotic dynamics in a singular chemotaxis system modeling onset of tumor angiogenesis, J. Differential Equations 260 (2016) 2225–2258. Crossref, Web of Science, Google Scholar
30. M. Winkler , Aggregation vs. global diffusive behavior in the higher-dimensional Keller–Segel model, J. Differential Equations 248 (2010) 2889–2905. Crossref, Web of Science, Google Scholar
31. M. Winkler , Global large-data solutions in a chemotaxis-(Navier–)Stokes system modeling cellular swimming in fluid drops, Comm. Partial Differential Equations 37 (2012) 319–351. Crossref, Web of Science, Google Scholar
32. 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
33. M. Winkler , Global weak solutions in a three-dimensional chemotaxis-Navier–Stokes system, Ann. Inst. H. Poincaré Anal. Non Linéaire 33 (2016) 1329–1352. Crossref, Web of Science, Google Scholar
34. M. Winkler , The two-dimensional Keller–Segel system with singular sensitivity and signal absorption: Global large-data solutions and their relaxation properties, Math. Models Methods Appl. Sci. 26 (2016) 987–1024. Link, Web of Science, Google Scholar
35. 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
36. M. Winkler , Global existence and stabilization in a degenerate chemotaxis-Stokes system with mildly strong diffusion enhancement, J. Differential Equations 264 (2018) 6109–6151. Crossref, Web of Science, Google Scholar
37. M. Winkler , Renormalized radial large-data solutions to the higher-dimensional Keller–Segel system with singular sensitivity and signal absorption, J. Differential Equations 264 (2018) 2310–2350. Crossref, Web of Science, Google Scholar
38. M. Winkler , Approaching logarithmic singularities in quasilinear chemotaxis-consumption systems with signal-dependent sensitivities, Discrete Contin. Dyn. Syst. Ser. B 27 (2022) 6565–6587. Crossref, Web of Science, Google Scholar
39. M. Winkler , Chemotaxis-Stokes interaction with very weak diffusion enhancement: Blow-up exclusion via detection of absorption-induced entropy structures involving multiplicative couplings, Adv. Nonlinear Stud. 22 (2022) 88–117. Crossref, Web of Science, Google Scholar
40. M. Winkler and K. C. Djie , Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect, Nonlinear Anal. 72 (2010) 1044–1064. Crossref, Web of Science, Google Scholar
41. J. Yan and Y. Li , Global generalized solutions to a Keller–Segel system with nonlinear diffusion and singular sensitivity, Nonlinear Anal. 176 (2018) 288–302. Crossref, Web of Science, Google Scholar
42. Q. Zhang and Y. Li , Convergence rates of solutions for a two-dimensional chemotaxis-Navier–Stokes system, Discrete Contin. Dyn. Syst. Ser. B 20 (2015) 2751–2759. Crossref, Web of Science, Google Scholar
43. Q. Zhang and Y. Li , Stabilization and convergence rate in a chemotaxis system with consumption of chemoattractant, J. Math. Phys. 56 (2015) 081506. Crossref, Web of Science, Google Scholar
44. X. Zhao and S. Zheng , Asymptotic behavior to a chemotaxis consumption system with singular sensitivity, Math. Methods Appl. Sci. 41 (2018) 2615–2624. Crossref, Web of Science, Google Scholar
45. X. Zhao and S. Zheng , Global existence and asymptotic behavior to a chemotaxis-consumption system with singular sensitivity and logistic source, Nonlinear Anal. Real World Appl. 42 (2018) 120–139. Crossref, Web of Science, Google Scholar