About DML-CZ | FAQ | Conditions of Use | Math Archives | Contact Us
  Previous |  Up |  Next

Article

Li, Yanjiang ; Yu, Zhongqing ; Huang, Yumei
Global classical solutions in a self-consistent chemotaxis(-Navier)-Stokes system. (English). Czechoslovak Mathematical Journal, vol. 74 (2024), issue 1, pp. 153-175
MSC: 35K55, 35Q35, 35Q92, 92C17 | MR 4717827 | Zbl 07893372 | DOI: 10.21136/CMJ.2023.0570-22
Full entry | Fulltext not available (moving wall 24 months)      Feedback
Keywords:
chemotaxis; Navier-Stokes system; self-consistent; global existence; boundedness
Summary:
The self-consistent chemotaxis-fluid system $$ \begin{cases} n_t+u\cdot \nabla n =\Delta n - \nabla \cdot (n\nabla c )+\nabla \cdot (n\nabla \phi ), &x\in \Omega ,\ t>0,\\ c_t +u\cdot \nabla c=\Delta c -nc,\quad &x\in \Omega ,\ t>0,\\ u_t+\kappa (u\cdot \nabla ) u+\nabla P=\Delta u - n\nabla \phi +n \nabla c,\qquad &x\in \Omega ,\ t>0,\\ \nabla \cdot u=0,\quad &x\in \Omega ,\ t>0, \end{cases} $$ is considered under no-flux boundary conditions for $n, c$ and the Dirichlet boundary condition for $u$ on a bounded smooth domain $ \Omega \subset \mathbb {R}^N$ $(N=2,3)$, $\kappa \in \lbrace 0,1 \rbrace $. The existence of global bounded classical solutions is proved under a smallness assumption on $\|c_{0}\|_{L^{\infty }(\Omega )}$. \endgraf Both the effect of gravity (potential force) on cells and the effect of the chemotactic force on fluid are considered here, and thus the coupling is stronger than the most studied chemotaxis-fluid systems. The literature on self-consistent chemotaxis-fluid systems of this type so far concentrates on the nonlinear cell diffusion as an additional dissipative mechanism. To the best of our knowledge, this is the first result on the boundedness of a self-consistent chemotaxis-fluid system with linear cell diffusion.
References:
[1] Bellomo, N., Bellouquid, A., Tao, Y., Winkler, 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] Bellomo, N., Outada, N., Soler, J., Tao, Y., Winkler, M.: Chemotaxis and cross-diffusion models in complex environments: Models and analytic problems toward a multiscale vision. Math. Models Methods Appl. Sci. 32 (2022), 713-792. DOI 10.1142/S0218202522500166 | MR 4421216 | Zbl 1497.35039
[3] Cao, X.: Global classical solutions in chemotaxis(-Navier)-Stokes system with rotational flux term. J. Differ. Equations 261 (2016), 6883-6914. DOI 10.1016/j.jde.2016.09.007 | MR 3562314 | Zbl 1352.35092
[4] Carrillo, J. A., Peng, Y., Xiang, Z.: Global existence and decay rates to self-consistent chemotaxis-fluid system. Available at https://arxiv.org/abs/2302.03274 (2023), 36 pages. MR 4671518
[5] Chae, M., Kang, K., Lee, J.: Existence of smooth solutions to coupled chemotaxis-fluid equations. Discrete Contin. Dyn. Syst. 33 (2013), 2271-2297. DOI 10.3934/dcds.2013.33.2271 | MR 3007686 | Zbl 1277.35276
[6] Chae, M., Kang, K., Lee, J.: Global existence and temporal decay in Keller-Segel models coupled to fluid equations. Commun. Partial Differ. Equations 39 (2014), 1205-1235. DOI 10.1080/03605302.2013.852224 | MR 3208807 | Zbl 1304.35481
[7] 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), 1437-1453. DOI 10.3934/dcds.2010.28.1437 | MR 2679718 | Zbl 1276.35103
[8] Duan, R., Li, X., Xiang, Z.: Global existence and large time behavior for a two-dimensional chemotaxis-Navier-Stokes system. J. Differ. Equations 263 (2017), 6284-6316. DOI 10.1016/j.jde.2017.07.015 | MR 3693175 | Zbl 1378.35160
[9] Duan, R., Lorz, A., Markowich, P. A.: Global solutions to the coupled chemotaxis-fluid equations. Commun. Partial Differ. Equations 35 (2010), 1635-1673. DOI 10.1080/03605302.2010.497199 | MR 2754058 | Zbl 1275.35005
[10] Galdi, G. P.: An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. I: Linearized Steady Problems. Springer Tracts in Natural Philosophy 38. Springer, Berlin (1994). DOI 10.1007/978-1-4757-3866-7 | MR 1284205 | Zbl 0949.35004
[11] He, P., Wang, Y., Zhao, L.: A further study on a 3D chemotaxis-Stokes system with tensor-valued sensitivity. Appl. Math. Lett. 90 (2019), 23-29. DOI 10.1016/j.aml.2018.09.019 | MR 3875493 | Zbl 1410.35249
[12] Horstmann, D.: From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. I. Jahresber. Dtsch. Math.-Ver. 105 (2003), 103-165. MR 2013508 | Zbl 1071.35001
[13] Ishida, S.: Global existence and boundedness for chemotaxis-Navier-Stokes systems with position-dependent sensitivity in 2D bounded domains. Discrete Contin. Dyn. Syst. 35 (2015), 3463-3482. DOI 10.3934/dcds.2015.35.3463 | MR 3320134 | Zbl 1308.35214
[14] Jin, C.: Global bounded solution in three-dimensional chemotaxis-Stokes model with arbitrary porous medium slow diffusion. Available at https://arxiv.org/abs/2101.11235 (2021), 24 pages. MR 4056454
[15] Jin, C., Wang, Y., Yin, J.: Global solvability and stability to a nutrient-taxis model with porous medium slow diffusion. Available at https://arxiv.org/abs/1804.03964 (2018), 35 pages. MR 4635735
[16] Keller, E. F., Segel, L. A.: Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol. 26 (1970), 399-415. DOI 10.1016/0022-5193(70)90092-5 | MR 3925816 | Zbl 1170.92306
[17] Kozono, H., Yanagisawa, T.: Leray's problem on the stationary Navier-Stokes equations with inhomogeneous boundary data. Math. Z. 262 (2009), 27-39. DOI 10.1007/s00209-008-0361-2 | MR 2491599 | Zbl 1169.35045
[18] Li, X., Wang, Y., Xiang, Z.: Global existence and boundedness in a 2D Keller-Segel-Stokes system with nonlinear diffusion and rotational flux. Commun. Math. Sci. 14 (2016), 1889-1910. DOI 10.4310/CMS.2016.v14.n7.a5 | MR 3549354 | Zbl 1351.35073
[19] Liu, J.-G., Lorz, A.: A coupled chemotaxis-fluid model: Global existence. Ann. Inst. H. Poincaré, Anal. Non Linéaire 28 (2011), 643-652. DOI 10.1016/J.ANIHPC.2011.04.005 | MR 2838394 | Zbl 1236.92013
[20] Lorz, A.: Coupled chemotaxis fluid model. Math. Models Methods Appl. Sci. 20 (2010), 987-1004. DOI 10.1142/S0218202510004507 | MR 2659745 | Zbl 1191.92004
[21] Patlak, C. S.: Random walk with persistence and external bias. Bull. Math. Biophys. 15 (1953), 311-338. DOI 10.1007/BF02476407 | MR 0081586 | Zbl 1296.82044
[22] Peng, Y., Xiang, Z.: Global existence and boundedness in a 3D Keller-Segel-Stokes system with nonlinear diffusion and rotational flux. Z. Angew. Math. Phys. 68 (2017), Article ID 68, 26 pages. DOI 10.1007/s00033-017-0816-6 | MR 3654908 | Zbl 1386.35189
[23] Tan, Z., Zhang, X.: Decay estimates of the coupled chemotaxis-fluid equations in $\Bbb{R}^3$. J. Math. Anal. Appl. 410 (2014), 27-38. DOI 10.1016/j.jmaa.2013.08.008 | MR 3109817 | Zbl 1333.92015
[24] Tao, Y.: Boundedness in a chemotaxis model with oxygen consumption by bacteria. J. Math. Anal. Appl. 381 (2011), 521-529. DOI 10.1016/j.jmaa.2011.02.041 | MR 2802089 | Zbl 1225.35118
[25] Tao, Y., Winkler, M.: Global existence and boundedness in a Keller-Segel-Stokes model with arbitrary porous medium diffusion. Discrete Contin. Dyn. Syst. 32 (2012), 1901-1914. DOI 10.3934/dcds.2012.32.1901 | MR 2871341 | Zbl 1276.35105
[26] Tuval, I., Cisneros, L., Dombrowski, C., Wolgemuth, C. W., Kessler, J. O., Goldstein, R. E.: Bacterial swimming and oxygen transport near contact lines. Proc. Natl. Acad. Sci. USA 102 (2005), 2277-2282. DOI 10.1073/pnas.0406724102 | Zbl 1277.35332
[27] 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
[28] 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
[29] Wang, Y., Winkler, M., Xiang, Z.: The small-convection limit in a two-dimensional chemotaxis-Navier-Stokes system. Math. Z. 289 (2018), 71-108. DOI 10.1007/s00209-017-1944-6 | MR 3803783 | Zbl 1397.35322
[30] 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
[31] Winkler, M.: Absence of collapse in a parabolic chemotaxis system with signal-dependent sensitivity. Math. Nachr. 283 (2010), 1664-1673. DOI 10.1002/mana.200810838 | MR 2759803 | Zbl 1205.35037
[32] 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
[33] Winkler, M.: Global large-data solutions in a chemotaxis-(Navier-)Stokes system modeling cellular swimming in fluid drops. Commun. Partial Differ. Equations 37 (2012), 319-351. DOI 10.1080/03605302.2011.591865 | MR 2876834 | Zbl 1236.35192
[34] Winkler, M.: Stabilization in a two-dimensional chemotaxis-Navier-Stokes system. Arch. Ration. Mech. Anal. 211 (2014), 455-487. DOI 10.1007/s00205-013-0678-9 | MR 3149063 | Zbl 1293.35220
[35] Winkler, M.: 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. DOI 10.1007/s00526-015-0922-2 | MR 3426095 | Zbl 1333.35104
[36] Winkler, M.: Global weak solutions in a three-dimensional chemotaxis-Navier-Stokes system. Ann. Inst. H. Poincaré, Anal. Non Linéaire 33 (2016), 1329-1352. DOI 10.1016/J.ANIHPC.2015.05.002 | MR 3542616 | Zbl 1351.35239
[37] Winkler, M.: How far do chemotaxis-driven forces influence regularity in the Navier- Stokes system?. Trans. Am. Math. Soc. 369 (2017), 3067-3125. DOI 10.1090/tran/6733 | MR 3605965 | Zbl 1356.35071
[38] Winkler, M.: Global existence and stabilization in a degenerate chemotaxis-Stokes system with mildly strong diffusion enhancement. J. Differ. Equations 264 (2018), 6109-6151. DOI 10.1016/j.jde.2018.01.027 | MR 3770046 | Zbl 1395.35038
[39] 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
[40] 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
[41] Zhang, Q., Li, Y.: Convergence rates of solutions for a two-dimensional chemotaxis- Navier-Stokes system. Discrete Contin. Dyn. Syst., Ser. B 20 (2015), 2751-2759. DOI 10.3934/dcdsb.2015.20.2751 | MR 3423254 | Zbl 1334.35104
[42] Zhang, Q., Li, Y.: Global weak solutions for the three-dimensional chemotaxis-Navier- Stokes system with nonlinear diffusion. J. Differ. Equations 259 (2015), 3730-3754. DOI 10.1016/j.jde.2015.05.012 | MR 3369260 | Zbl 1320.35352
[43] Zhang, Q., Zheng, X.: Global well-posedness for the two-dimensional incompressible chemotaxis-Navier-Stokes equations. SIAM J. Math. Anal. 46 (2014), 3078-3105. DOI 10.1137/130936920 | MR 3252810 | Zbl 1444.35011
[44] Zheng, J.: Global existence and boundedness in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion and general sensitivity. Ann. Mat. Pura Appl. (4) 201 (2022), 243-288. DOI 10.1007/s10231-021-01115-4 | MR 4375009 | Zbl 1485.35119
Partner of
EuDML logo