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Keywords:
chemotaxis; self-consistent; weak solution; consumption
chemotaxis; self-consistent; weak solution; consumption
Summary:
We study the self-consistent chemotaxis-fluid system with nonlinear resource consumption $$ \begin {cases} n_{t}+u\cdot \nabla n=\Delta n^m -\nabla \cdot (n \nabla c)+\nabla \cdot (n\nabla \phi ), & x\in \Omega ,\ t>0, \\ c_{t}+u\cdot \nabla c=\Delta c-n^\alpha c, & x\in \Omega ,\ t>0, \\ u_t+ \nabla P=\Delta u-n\nabla \phi +n \nabla c,& x\in \Omega ,\ t>0,\\ \nabla \cdot u=0,& x\in \Omega ,\ t>0,\\ \end {cases} $$ under no-flux boundary conditions in a bounded domain $\Omega \subset \mathbb {R}^3$ with smooth boundary. It is proved that this system possesses a global weak solution provided $m>1$ and $\alpha > \frac {4}{3}$.
References:
[1] Bellomo, N., Bellouquid, A., Tao, Y., Winker, M.: Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues. Math. Models Methods Appl. Sci. 25 (2015), 1663-1763. DOI 10.1142/S021820251550044X | MR 3351175 | Zbl 1326.35397
[2] Francesco, M. Di, Lorz, A., Markowich, P. A.: Chemotaxis-fluid coupled model for swimming bacteria with nonlinear diffusion: Global existence and asymptotic behavior. Discrete Contin. Dyn. Syst. 28 (2010), 1437-1452. DOI 10.3934/dcds.2010.28.1437 | MR 2679718 | Zbl 1276.35103
[3] Keller, E. F., Segel, L. A.: Model for chemotaxis. J. Theor. Biol. 30 (1971), 225-234. DOI 10.1016/0022-5193(71)90050-6 | Zbl 1170.92307
[4] Khelghati, A., Baghaei, K.: Boundedness of classical solutions for a chemotaxis model with rotational flux terms. ZAMM, Z. Angew. Math. Mech. 98 (2018), 1864-1877. DOI 10.1002/zamm.201700091 | MR 3864015 | Zbl 07776936
[5] Liu, Y., Zhuang, Y.: Boundedness in a high-dimensional forager-exploiter model with nonlinear resource consumption by two species. Z. Angew. Math. Phys. 71 (2020), Article ID 151, 18 pages. DOI 10.1007/s00033-020-01376-8 | MR 4141610 | Zbl 1448.35292
[6] Mizoguchi, N., Souplet, P.: Nondegeneracy of blow-up points for the parabolic Keller-Segel system. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 31 (2014), 851-875. DOI 10.1016/J.ANIHPC.2013.07.007 | MR 3249815 | Zbl 1302.35075
[7] Nagai, T.: Blow-up of radially symmetric solutions to a chemotaxis system. Adv. Math. Sci. Appl. 5 (1995), 581-601. MR 1361006 | Zbl 0843.92007
[8] Nam, K.-M., Li, K.-O., Kim, Y.-H.: Boundedness of solutions to a 2D chemotaxis-Navier-Stokes system with general sensitivity and nonlinear diffusion. Nonlinear Anal., Real World Appl. 73 (2023), Article ID 103906, 15 pages. DOI 10.1016/j.nonrwa.2023.103906 | MR 4575115 | Zbl 1519.35226
[9] Ou, H., Wang, L.: Boundedness in a two-species chemotaxis system with nonlinear resource consumption. Qual. Theory Dyn. Syst. 23 (2024), Article ID 14, 16 pages. DOI 10.1007/s12346-023-00873-1 | MR 4648741 | Zbl 1527.92013
[10] Sohr, H.: The Navier-Stokes Equations: An Elementary Functional Analytic Approach. Birkhäuser, Basel (2001). DOI 10.1007/978-3-0348-8255-2 | MR 3013225 | Zbl 0983.35004
[11] Wang, Y.: Global solvability in a two-dimensional self-consistent chemotaxis-Navier-Stokes system. Discrete Contin. Dyn. Syst., Ser. S 13 (2020), 329-349. DOI 10.3934/dcdss.2020019 | MR 4043697 | Zbl 1442.35481
[12] Wang, Y., Winkler, M., Xiang, Z.: Global classical solutions in a two-dimensional chemotaxis-Navier-Stokes system with subcritical sensitivity. Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 18 (2018), 421-466. DOI 10.2422/2036-2145.201603_004 | MR 3801284 | Zbl 1395.92024
[13] Wang, Y., Zhao, L.: A 3D self-consistent chemotaxis-fluid system with nonlinear diffusion. J. Differ. Equations 269 (2020), 148-179. DOI 10.1016/j.jde.2019.12.002 | MR 4081519 | Zbl 1436.35238
[14] Winkler, M.: Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model. J. Differ. Equations 248 (2010), 2889-2905. DOI 10.1016/j.jde.2010.02.008 | MR 2644137 | Zbl 1190.92004
[15] Winkler, M.: A three-dimensional Keller-Segel-Navier-Stokes system with logistic source: Global weak solutions and asymptotic stabilization. J. Funct. Anal. 276 (2019), 1339-1401. DOI 10.1016/j.jfa.2018.12.009 | MR 3912779 | Zbl 1408.35132
[16] Wu, D., Shen, S.: Global boundedness and stabilization in a forager-exploiter model with logistic growth and nonlinear resource consumption. Nonlinear Anal., Real World Appl. 72 (2023), Article ID 103854, 19 pages. DOI 10.1016/j.nonrwa.2023.103854 | MR 4548493 | Zbl 1517.35053
[17] Xie, J., Zheng, J.: A new result for global existence and boundedness in a three-dimensional self-consistent chemotaxis-fluid system with nonlinear diffusion. Acta Appl. Math. 183 (2023), Article ID 5, 36 pages. DOI 10.1007/s10440-022-00552-4 | MR 4534454 | Zbl 1507.35041
[18] Yu, P.: Global existence and boundedness in a chemotaxis-Stokes system with arbitrary porous medium diffusion. Math. Methods Appl. Sci. 43 (2020), 639-657. DOI 10.1002/mma.5920 | MR 4056454 | Zbl 1439.35254
[19] Zhang, Q., Tao, W.: Boundedness and stabilization in a two-species chemotaxis system with signal absorption. Comput. Math. Appl. 78 (2019), 2672-2681. DOI 10.1016/j.camwa.2019.04.008 | MR 4001731 | Zbl 1443.92062
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