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The convergence rate of the fast signal diffusion limit for a Keller–Segel–Stokes system with large initial data

Part of: Physiological, cellular and medical topics Equations of mathematical physics and other areas of application Parabolic equations and systems

Published online by Cambridge University Press:  23 December 2020

Min Li  and
Zhaoyin Xiang
Min Li
Affiliation:
School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, 611731, China ([email protected]; [email protected])
Zhaoyin Xiang
Affiliation:
School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, 611731, China ([email protected]; [email protected])
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Abstract

In this paper, we investigate the fast signal diffusion limit of solutions of the fully parabolic Keller–Segel–Stokes system to solution of the parabolic–elliptic-fluid counterpart in a two-dimensional or three-dimensional bounded domain with smooth boundary. Under the natural volume-filling assumption, we establish an algebraic convergence rate of the fast signal diffusion limit for general large initial data by developing a series of subtle bootstrap arguments for combinational functionals and using some maximal regularities. In our current setting, in particular, we can remove the restriction to asserting convergence only along some subsequence in Wang–Winkler and the second author (Cal. Var., 2019).

Keywords

Keller–Segel–Stokes systemlarge initial datafast signal diffusion limitconvergence rate

MSC classification

Primary: 35K55: Nonlinear parabolic equations
Secondary: 35Q92: PDEs in connection with biology and other natural sciences 92C17: Cell movement (chemotaxis, etc.)
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 151 , Issue 6 , December 2021 , pp. 1972 - 2012
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Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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References

Biler, P. and Brandolese, L.. On the parabolic–elliptic limit of the doubly parabolic Keller–Segel system modelling chemotaxis. Stud. Math. 193 (2009), 241–261.CrossRefGoogle Scholar
Black, T.. The Stokes limit in a three-dimensional chemotaxis-Navier–Stokes system. J. Math. Fluid Mech. 22 (2020), 35.CrossRefGoogle Scholar
Blanchet, A., Carrillo, J. A. and Masmoudi, N.. Infinite time aggregation for the critical Patlak–Keller–Segel model in ${\mathbb{R}} ^2$. Comm. Pure Appl. Math. 61 (2008), 1449–1481.CrossRefGoogle Scholar
Cao, X. and Lankeit, J.. Global classical small-data solutions for a three-dimensional chemotaxis Navier–Stokes system involving matrix-valued sensitivities. Calc. Var. 55 (2016), 107.CrossRefGoogle Scholar
Duan, R., Lorz, A. and Markowich, P. A.. Global solutions to the coupled chemotaxis–fluid equations. Comm. Part. Diff. Eqs. 35 (2010), 1635–1673.CrossRefGoogle Scholar
Evans, L. C.. Partial Differential Equations, 2nd edn (Providence: American Mathematical Society, 2010).Google Scholar
Freitag, M.. The fast signal diffusion limit in nonlinear chemotaxis systems. Disc. Cont. Dyn. Syt. -B 25 (2020), 1109–1128.Google Scholar
Giga, Y. and Sohr, H.. 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.CrossRefGoogle Scholar
Ghoul, T. and Masmoudi, N.. Minimal mass blowup solutions for the Patlak–Keller–Segel equation. Comm. Pure Appl. Math. 71 (2018), 1957–2015.CrossRefGoogle Scholar
Herrero, M. A. and Velázquez, J. J. L.. A blow-up mechanism for a chemotaxis model, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 24 (1997), 633–683.Google Scholar
Horstmann, D. and Winkler, M.. Boundedness vs. blow-up in a chemotaxis system. J. Diff. Equ. 215 (2005), 52–107.CrossRefGoogle Scholar
Keller, E. F. and Segel, L. A.. Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol. 26 (1970), 399–415.CrossRefGoogle ScholarPubMed
Kurokiba, M. and Ogawa, T.. Singular limit problem for the Keller–Segel system and drift-diffusion system in scaling critical spaces. J. Evol. Equ. 20 (2020), 421–457.CrossRefGoogle Scholar
Lemarié-Rieusset, P. G.. Small data in an optimal Banach space for the parabolic–parabolic and parabolic–elliptic Keller–Segel equations in the whole space. Adv. Differential Equations 18 (2013), 1189–1208.Google Scholar
Liu, J., Wang, L. and Zhou, Z.. Positivity-preserving and asymptotic preserving method for 2D Keller–Segal equations. Mathematics of Computation 87 (2018), 1165–1189.CrossRefGoogle Scholar
Lorz, A.. Coupled chemotaxis fluid model. Math. Mod. Meth. Appl. Sci. 20 (2012), 987–1004.CrossRefGoogle Scholar
Lorz, A.. A coupled Keller–Segel–Stokes model: global existence for small initial data and blow-up delay. Commun. Math. Sci. 10 (2012), 555–574.CrossRefGoogle Scholar
Mizoguchi, N. and Souplet, P.. Nondegeneracy of blow-up points for the parabolic Keller–Segel system. Ann. I. H. Poincaré - AN 31 (2014), 851–875.CrossRefGoogle Scholar
Mizukami, M.. The fast signal diffusion limit in a Keller–Segel system. J. Math. Anal. Appl. 472 (2019), 1313–1330.CrossRefGoogle Scholar
Nagai, T., Senba, T. and Yoshida, K.. Applicationuio of the Trudinger–Moser inequality to a parabolic system of chemotaxis. Funkcial. Ekvac. 40 (1997), 411–433.Google Scholar
Osaki, K. and Yasgi, A.. Finite dimensional attractor for one-dimensional Keller–Segel equations. Funkcial. Ekvac. 44 (2001), 441–469.Google Scholar
Peng, Y. and Xiang, Z.. Global existence and convergence rates to a chemotaxis–fluids system with mixed boundary conditions. J. Differential Equations 267 (2019), 1277–1321.CrossRefGoogle Scholar
Raczynski, A.. Stability property of the two-dimensional Keller–Segel model. Asympt. Anal. 61 (2009), 35–59.Google Scholar
Senba, T. and Suzuki, T.. Chemotactic collapse in a parabolic–elliptic system of mathematical biology. Adv. Diff. Equ. 6 (2001), 21–50.Google Scholar
Tian, Y. and Xiang, Z.. Global solutions to a 3D chemotaxis-Stokes system with nonlinear cell diffusion and Robin signal boundary condition. J. Diff. Equ. 269 (2020), 2012–2056.CrossRefGoogle Scholar
Tuval, I., Cisneros, L., Dombrowski, C., Wolgemuth, C. W., Kessler, J. O. and Goldstein, R. E.. Bacterial swimming and oxygen transport near contact lines. Proc. Nat. Acad. Sci. USA. 102 (2005), 2277–2282.CrossRefGoogle ScholarPubMed
Wang, Y.. Global weak solutions in a three-dimensional Keller–Segel–Navier–Stokes system with subcritical sensitivity. Math. Models Methods Appl. Sci. 27 (2017), 2745–2780.CrossRefGoogle Scholar
Wang, Y., Winkler, M. and Xiang, Z.. Global classical solutions in a two-dimensional chemotaxis-Navier–Stokes system with subcritical sensitivity. Ann. Sc. Norm. Super. Pisa Cl. Sci. XVIII (2018), 421–466.Google Scholar
Wang, Y., Winkler, M. and Xiang, Z.. The small-convection limit in a two-dimensional chemotaxis-Navier–Stokes system. Math. Z. 289 (2018), 71–108.CrossRefGoogle Scholar
Wang, Y., Winkler, M. and Xiang, Z.. The fast signal diffusion limit in Keller–Segel(-fluid) systems. Calc. Var. 58 (2019), 40.CrossRefGoogle Scholar
Wang, Y. and Xiang, Z.. Global existence and boundedness in a Keller–Segel–Stokes system involving a tensor-valued sensitivity with saturation. J. Diff. Equ. 259 (2015), 7578–7609.CrossRefGoogle Scholar
Wang, Y. and Xiang, Z.. Global existence and boundedness in a Keller–Segel–Stokes system involving a tensor-valued sensitivity with saturation: the 3D case. J. Diff. Equ. 261 (2016), 4944–4973.CrossRefGoogle Scholar
Winkler, M.. Global large-data solutions in a chemotaxis-(Navier–)Stokes system modeling cellular swimming in fluid drops. Comm. Part. Diff. Eqs. 37 (2012), 319–351.CrossRefGoogle Scholar
Winkler, M.. Finite-time blow-up in the higher-dimensional parabolic–parabolic Keller–Segel system. J. Math. Pures. Appl. 100 (2013), 748–767.CrossRefGoogle Scholar
Winkler, M.. Stabilization in a two-dimensional chemotaxis-Navier–Stokes system. Arch. Ration. Mech. Anal. 211 (2014), 455–487.CrossRefGoogle Scholar
Winkler, M.. Global weak solutions in a three-dimensional chemotaxis-Navier–Stokes system. Ann. I. H. Poincaré-AN 33 (2016), 1329–1352.CrossRefGoogle Scholar
Winkler, M.. How far do chemotaxis-driven forces influence regularity in the Navier–Stokes system?. Trans. Amer. Math. Soc. 369 (2017), 3067–3125.CrossRefGoogle Scholar
Winkler, M.. Does fluid interaction affect regularity in the three-dimensional Keller–Segel system with saturated sensitivity?. J. Math. Fluid Mech 20 (2018), 1889–1909.CrossRefGoogle Scholar
Wu, C. and Xiang, Z.. The small-convection limit in a two-dimensional Keller–Segel–Navier–Stokes system. J. Diff. Equ. 267 (2019), 938–978.CrossRefGoogle Scholar
Wu, C. and Xiang, Z.. Asymptotic dynamics on a chemotaxis-Navier–Stokes system with nonlinear diffusion and inhomogeneous boundary conditions. Math. Models Methods Appl. Sci. 30 (2020), 1325–1374.CrossRefGoogle Scholar
Xue, C. and Othmer, H. G.. Multiscale models of taxis-driven patterning in bacterial populations. SIAM J. Appl. Math. 70 (2009), 133–169.CrossRefGoogle ScholarPubMed