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Global solvability and eventual smoothness in a chemotaxis-fluid system with weak logistic-type degradation

Yulan Wang

School of Science, Xihua University, Chengdu 610039, P. R. China

https://doi.org/10.1142/S0218202520400102Cited by:18 (Source: Crossref)

This article is part of the issue:

Special Issue on "Chemotaxis Systems in Complex Frameworks:Pattern Formation, Qualitative Analysis and Blowup Prevention"
Guest Editors: N. Bellomo, Y. Tao and M. Winkler

Abstract

We consider the coupled chemotaxis–Navier–Stokes system with logistic source term

in a bounded, smooth domain

$Ω \subset ℝ^{3}$ , where

$Φ \in W^{2, \infty} (Ω)$ and where

$r \geq 0$ ,

$μ > 0$ and

$1 < α < 2$ are given parameters. Although the degradation here is weaker than the usual quadratic case, it is proved that for any sufficiently regular initial data, the initial-value problem for this system under no-flux boundary conditions for

$n$ and

$c$ and homogeneous Dirichlet boundary condition for

$u$ possesses at least one globally defined weak solution. And this weak solution becomes smooth after some waiting time provided

$\frac{6}{5} \leq α < 2$ .

Communicated by N. Bellomo, Y. Tao and M. Winkler

Keywords:

AMSC: 92C17, 35Q30, 35K55, 35B65, 35B40

References

1. H. Amann, Compact embedding of vector-valued Sobolev and Beseov space, Glas. Mat. Ser. III 35 (2000) 161–177. Google Scholar
2. K. Baghaeia and A. Khelghatib, Global existence and boundedness of classical solutions for a chemotaxis model with consumption of chemoattractant and logistic source, Math. Methods Appl. Sci. 40 (2017) 3799–3807. Crossref, Web of Science, Google Scholar
3. N. Bellomo, A. Bellouquid and N. Chouhad, From a multiscale derivation of nonlinear cross-diffusion models to Keller–Segel models in a Navier–Stokes fluid, Math. Mod. Methods Appl. Sci. 26 (2016) 2041–2069. Link, Web of Science, Google Scholar
4. N. Bellomo, A. Bellouquid, Y. Tao and M. Winkler, Toward a mathematical theory of Keller–Segel models of pattern formation in biological tissues, Math. Mod. Methods Appl. Sci. 25 (2015) 1663–1763. Link, Web of Science, Google Scholar
5. T. Black, The Stokes limit in a three-dimensional chemotaxis-Navier–Stokes system, J. Math. Fluid Mech. 22 (2020) 1. Crossref, Web of Science, Google Scholar
6. M. Chae, K. Kang and J. Lee, Global existence and temporal decay in Keller-Segel models coupled to fluid equations, Comm. Part. Differ. Eq. 39 (2014) 1205–1235. Crossref, Web of Science, Google Scholar
7. E. S. Citlanadze, On the variational theory of a class of non-linear operators in the space $L_{p}$ ( $p > 1$ ) (Russian), Doklady Akad. Nauk SSSR (N.S.) 71 (1950) 441–444. Google Scholar
8. R. Duan, A. Lorz and P. A. Markowich, Global solutions to the coupled chemotaxis-fluid equations, Comm. Part. Differ. Eqs. 35 (2010) 1635–1673. Crossref, Web of Science, 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, 1981). Crossref, Google Scholar
12. M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Scuola Normale Superiore Pisa 24 (1997) 633–683. Google Scholar
13. H. Jin and T. Xiang, Chemotaxis effect versus logistic damping on boundedness for the 2-D minimal Keller–Segel model, C. R. Acad. Sci. Paris Ser. I 356 (2018) 875–885. Crossref, Web of Science, Google Scholar
14. O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural’ceva, Linear and Quasi-Linear Equations of Parabolic Type, American Mathematical Society Translation, Vol. 23 (Providence, 1968). Crossref, Google Scholar
15. J. Lankeit, Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source, J. Differential Equations 258 (2015) 1158–1191. Crossref, Web of Science, Google Scholar
16. J. Lankeit, Long-term behavior in a chemotaxis-fluid system with logistic source, Math. Models Methods Appl. Sci. 26 (2016) 2071–2109. Link, Web of Science, Google Scholar
17. J. Lankeit and Y. Wang, Global existence, boundedness and stabilization in a highdimensional chemotaxis system with consumption, Discrete Contin. Dyn. Syst. 37 (2017) 6099–6121. Crossref, Web of Science, Google Scholar
18. D. S. Mitrinović and P. M. Vasić, Analytic Inequalities (Springer-Verlag, 1970). Crossref, Google Scholar
19. M. Mizukami, How strongly does diffusion or logistic-type degradation affect existence of global weak solutions in a chemotaxis-Navier-Stokes system?, preprint (2018), arXiv:1810.01098v1. Google Scholar
20. N. Mizoguchi and M. Winkler, Blow-up in the two-dimensional parabolic Keller-Segel system, preprint. Google Scholar
21. K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura, Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Anal. 51 (2002) 119–144. Crossref, Web of Science, Google Scholar
22. F. J. Richards, A flexible growth function for empirical use, J. Exp. Botany 10 (1959) 290–300. Crossref, Web of Science, Google Scholar
23. H. Sohr, The Navier–Stokes Equations: An Elementary Functional Analytic Approach (Birkhäuser, 2001). Crossref, Google Scholar
24. 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
25. Y. Tao and M. Winkler, Eventual smoothness and stabilization of large-data solutions in a three-dimensional chemotaxis system with consumption of chemoattractant, J. Differential Equations 252 (2012) 2520–2543. Crossref, Web of Science, Google Scholar
26. Y. Tao and M. Winkler, Boundedness and decay enforced by quadratic degradation in a three-dimensional chemotaxis-fluid system, Z. Angew. Math. Phys. 66 (2015) 2555–2573. Crossref, Web of Science, Google Scholar
27. Y. Tao and M. Winkler, Blow-up prevention by quadratic degradation in a two-dimensional Keller–Segel-Navier–Stokes system, Z. Angew. Math. Phys. 67 (2016) 138. Crossref, Web of Science, Google Scholar
28. J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Comm. Part. Differ. Eq. 32 (2007) 849–877. Crossref, Web of Science, Google Scholar
29. 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. Nat. Acad. Sci. USA 102 (2005) 2277–2282. Crossref, Web of Science, Google Scholar
30. P. F. Verhulst, Notice sur la loi que la population suit dans son accroissement, Curr. Math. Phys. 10 (1838) 113–126. Google Scholar
31. G. Viglialoro, Very weak global solutions to a parabolic–parabolic chemotaxis-system with logistic source, J. Math. Anal. Appl. 439 (2016) 197–212. Crossref, Web of Science, Google Scholar
32. L. Von Bertalanffy, A quantitative theory of organic growth, Human Biol. 10 (1938) 181–213. Google Scholar
33. D. Vorotnikov, Weak solutions for a bioconvection model related to Bacillus subtilis, Commun. Math. Sci. 12 (2014) 545–563. Crossref, Web of Science, Google Scholar
34. 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. Crossref, Web of Science, Google Scholar
35. M. Winkler, Chemotaxis with logistic source: Very weak global solutions and boundedness properties, J. Math. Anal. Appl. 348 (2008) 708–729. Crossref, Web of Science, Google Scholar
36. M. Winkler, Boundedness in the higher-dimensional parabolic–parabolic chemotaxis system with logistic source, Comm. Part. Differ. Eqs. 35 (2010) 1516–1537. Crossref, Web of Science, Google Scholar
37. M. Winkler, Global large-data solutions in a chemotaxis-(Navier-)Stokes system modeling cellular swimming in fluid drops, Comm. Partial Differ. Eqs. 37 (2012) 319–351. Crossref, Web of Science, Google Scholar
38. M. Winkler, Finite-time blow-up in the higher-dimensional parabolic–parabolic Keller–Segel system, J. Math. Pures Appl. 100 (2013) 748–767. Crossref, Web of Science, Google Scholar
39. 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
40. M. Winkler, Global weak solutions in a three-dimensional chemotaxis-Navier–Stokes system, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 33 (2016) 1329–1352. Crossref, Web of Science, Google Scholar
41. 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
42. M. Winkler, A three-dimensional Keller–Segel–Navier–Stokes system with logistic source: Global weak solutions and asymptotic stabilization, J. Funct. Anal. 276 (2019) 1339–1401. Crossref, Web of Science, Google Scholar
43. M. Winkler, The role of superlinear damping in the construction of solutions to drift-diffusion problems with initial data in $L^{1}$ , Adv. Nonlinear Anal. 9 (2020) 526–566. Crossref, Web of Science, Google Scholar
44. T. Xiang, Boundedness and global existence in the higher-dimensional parabolic–parabolic chemotaxis system with/without growth source, J. Differential Equations 258 (2015) 4275–4323. Crossref, Web of Science, Google Scholar
45. Q. Zhang and Y. Li, Global weak solutions for the three-dimensional chemotaxis-Navier–Stokes system with nonlinear diffusion, J. Differential Equations 259 (2015) 3730–3754. 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
47. Q. Zhang and X. Zheng, Global well-posedness for the two-dimensional incompressible chemotaxis-Navier–Stokes equations, SIAM J. Math. Anal. 46 (2014) 3078–3105. Crossref, Web of Science, Google Scholar