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Blow-up prevention by saturated chemotactic sensitivity in a 2D Keller-Segel-Stokes system. (English) Zbl 1470.35185

In this paper, the author consider the global existence of classical solutions to a chemotaxis-Stokes system with indirect signal production and saturated chemotaxis sensitivity in a bounded domain of \(\mathbb R^2\). This is a nature extension of the chemotaxis model with indirect signal production. Using energy method, the author establish the existence of global classical solution.

MSC:

35K51 Initial-boundary value problems for second-order parabolic systems
35K59 Quasilinear parabolic equations
76D07 Stokes and related (Oseen, etc.) flows
92C17 Cell movement (chemotaxis, etc.)
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[1] Bellomo, N.; Bellouquid, A.; Chouhad, N., From a multiscale derivation of nonlinear cross-diffusion models to Keller-Segel models in a Navier-Stokes fluid, Math. Models Methods Appl. Sci., 26, 11, 2041-2069 (2016) · Zbl 1353.35038
[2] Cao, X.; Lankeit, J., Global classical small-data solutions for a three-dimensional chemotaxis Navier-Stokes system involving matrix-valued sensitivities, Calc. Var. Partial Differ. Equ., 55, 4, 107-146 (2016) · Zbl 1366.35075
[3] Chae, M.; Kang, K.; Lee, J., Existence of smooth solutions to coupled chemotaxis-fluid equations, Discrete Contin. Dyn. Syst., Ser. A, 33, 6, 2271-2297 (2013) · Zbl 1277.35276
[4] Chae, M.; Kang, K.; Lee, J., Global existence and temporal decay in Keller-Segel models coupled to fluid equations, Commun. Partial Differ. Equ., 39, 7, 1205-1235 (2014) · Zbl 1304.35481
[5] Duan, R.; Li, X.; Xiang, Z., Global existence and large time behavior for a two dimensional chemotaxis-Navier-Stokes system, J. Differ. Equ., 263, 10, 6284-6316 (2017) · Zbl 1378.35160
[6] Duan, R.; Xiang, Z., A note on global existence for the chemotaxis-Stokes model with nonlinear diffusion, Int. Math. Res. Not., 2014, 7, 1833-1852 (2014) · Zbl 1323.35184
[7] Espejo, E.; Winkler, M., Global classical solvability and stabilization in a two-dimensional chemotaxis-Navier-Stokes system modeling coral fertilization, Nonlinearity, 31, 4, 1227-1259 (2018) · Zbl 1392.35042
[8] Fujie, K.; Senba, T., Application of an Adams type inequality to a two-chemical substances chemotaxis system, J. Differ. Equ., 263, 7, 88-148 (2017) · Zbl 1364.35120
[9] Fujie, K.; Senba, T., Blowup of solutions to a two-chemical substances chemotaxis system in the critical dimension, J. Differ. Equ., 266, 7, 942-976 (2019) · Zbl 1406.35149
[10] Hillen, T.; Painter, K. J., A user’s guide to PDE models in a chemotaxis, J. Math. Biol., 58, 183-217 (2009) · Zbl 1161.92003
[11] Horstmann, D., From 1970 until present: the Keller-Segel model in chemotaxis and its consequences. I, Jahresber. Dtsch. Math.-Ver., 105, 103-165 (2003) · Zbl 1071.35001
[12] Horstmann, D.; Wang, G., Blow-up in a chemotaxis model without symmetry assumptions, Eur. J. Appl. Math., 12, 2, 159-177 (2001) · Zbl 1017.92006
[13] Horstmann, D.; Winkler, M., Boundedness vs. blow-up in a chemotaxis system, J. Differ. Equ., 215, 1, 52-107 (2005) · Zbl 1085.35065
[14] Hu, B.; Tao, Y., To the exclusion of blow-up in a three-dimensional chemotaxis-growth model with indirect attractant production, Math. Models Methods Appl. Sci., 26, 11, 2111-2128 (2016) · Zbl 1351.35076
[15] Ishida, S.; Seki, K.; Yokota, T., Boundedness in quasilinear Keller-Segel systems of parabolic-parabolic type on non-convex bounded domains, J. Differ. Equ., 256, 8, 2993-3010 (2014) · Zbl 1295.35252
[16] Keller, E. F.; Segel, L. A., Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26, 8, 399-415 (1970) · Zbl 1170.92306
[17] Kiselev, A.; Xu, X., Suppression of chemotactic explosion by mixing, Arch. Ration. Mech. Anal., 222, 2, 1077-1112 (2016) · Zbl 1351.35233
[18] Kozono, H.; Miura, M.; Sugiyama, Y., Existence and uniqueness theorem on mild solutions to the Keller-Segel system coupled with the Navier-Stokes fluid, J. Funct. Anal., 270, 5, 1663-1683 (2016) · Zbl 1343.35069
[19] Ladyzhenskaia, O. A.; Solonnikov, V. A.; Uralćeva, N. N., Linear and Quasi-Linear Equations of Parabolic Type (1968), St. Petersburg: Russian Academy of Sciences, St. Petersburg · Zbl 0174.15403
[20] Li, H.; Tao, Y., Boundedness in a chemotaxis system with indirect signal production and generalized logistic source, Appl. Math. Lett., 1, 77, 108-113 (2018) · Zbl 1503.92021
[21] Li, T.; Suen, A.; Winkler, M.; Xue, C., Small-data solutions in a chemotaxis system with rotation, Math. Models Methods Appl. Sci., 25, 3, 721-747 (2015) · Zbl 1322.35054
[22] Lorz, A., A coupled Keller-Segel-Stokes model: global existence for small initial data and blow-up delay, Commun. Math. Sci., 10, 2, 555-574 (2012) · Zbl 1282.35138
[23] Nagai, T.; Senba, T.; Yoshida, K., Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkc. Ekvacioj, 40, 411-433 (1997) · Zbl 0901.35104
[24] Nanjundiah, V., Chemotaxis, signal relaying and aggregation morphology, J. Theor. Biol., 42, 1, 63-105 (1973)
[25] Osaki, K.; Yagi, A., Finite dimensional attractor for one-dimensional Keller-Segel equations, Funkc. Ekvacioj, 44, 441-469 (2001) · Zbl 1145.37337
[26] Painter, K.; Hillen, T., Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q., 10, 4, 501-543 (2002) · Zbl 1057.92013
[27] Pedley, T.; Kessler, J., Hydrodynamic phenomena in suspensions of swimming microorganisms, Annu. Rev. Fluid Mech., 24, 1, 313-358 (1992) · Zbl 0825.76985
[28] Peng, Y.; Xiang, Z., Global solutions to the coupled chemotaxis-fluids system in a 3D unbounded domain with boundary, Math. Models Methods Appl. Sci., 28, 869-920 (2018) · Zbl 1391.35206
[29] Peng, Y.; Xiang, Z., Global existence and convergence rates to a chemotaxis-fluids system with mixed boundary conditions, J. Differ. Equ., 267, 2, 1277-1321 (2019) · Zbl 1412.35174
[30] Porzio, M.; Vespri, V., Hölder estimates for local solutions of some doubly nonlinear degenerate parabolic equations, J. Differ. Equ., 103, 1, 146-178 (1993) · Zbl 0796.35089
[31] Romanczuk, P.; Erdmann, U.; Engel, H., Beyond the Keller-Segel model, Eur. Phys. J. Spec. Top., 157, 1, 61-77 (2008)
[32] Strohm, S.; Tyson, R. C.; Powell, J. A., Pattern formation in a model for mountain pine beetle dispersal: linking model predictions to data, Bull. Math. Biol., 75, 10, 1778-1797 (2013) · Zbl 1275.92006
[33] Tao, Y.; Winkler, M., Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differ. Equ., 252, 1, 692-715 (2012) · Zbl 1382.35127
[34] Tao, T.; Winkler, M., Critical mass for infinite-time aggregation in a chemotaxis model with indirect signal production, J. Eur. Math. Soc., 19, 12, 3641-3678 (2017) · Zbl 1406.35068
[35] Tuval, I.; Cisneros, L.; Dombrowski, C., Bacterial swimming and oxygen transport near contact lines, Proc. Natl. Acad. Sci., 102, 7, 2277-2282 (2005) · Zbl 1277.35332
[36] Wang, Y., Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system with subcritical sensitivity, Math. Models Methods Appl. Sci., 27, 2745-2780 (2017) · Zbl 1378.92010
[37] Wang, Y.; Winkler, M.; Xiang, Z., The small-convection limit in a two-dimensional chemotaxis-Navier-Stokes system, Math. Z., 289, 1-2, 71-108 (2018) · Zbl 1397.35322
[38] 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., 18, 2, 421-466 (2018) · Zbl 1395.92024
[39] Wang, Y.; Winkler, M.; Xiang, Z., The fast signal diffusion limit in Keller-Segel(-fluid) systems, Calc. Var., 58, 196 (2019) · Zbl 1426.92009
[40] Wang, Y.; Xiang, Z., Global existence and boundedness in a Keller-Segel-Stokes system involving a tensor-valued sensitivity with saturation, J. Differ. Equ., 259, 2, 7578-7609 (2015) · Zbl 1323.35071
[41] Wang, Y.; Xiang, Z., Global existence and boundedness in a Keller-Segel-Stokes system involving a tensor-valued sensitivity with saturation: the 3D case, J. Differ. Equ., 261, 12, 4944-4973 (2016) · Zbl 1345.35054
[42] Winkler, M., Global large-data solutions in a chemotaxis-(Navier-)Stokes system modeling cellular swimming in fluid drops, Commun. Partial Differ. Equ., 37, 2, 319-351 (2012) · Zbl 1236.35192
[43] Winkler, M., Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100, 5, 748-767 (2013) · Zbl 1326.35053
[44] Winkler, M., Chemotactic cross-diffusion in complex frameworks, Math. Models Methods Appl. Sci., 26, 11, 2035-2040 (2016) · Zbl 1515.35056
[45] Winkler, M., A three-dimensional Keller-Segel-Navier-Stokes system with logistic source: global weak solutions and asymptotic stabilization, J. Funct. Anal., 276, 11, 1339-1401 (2019) · Zbl 1408.35132
[46] Wu, C.; Xiang, Z., The small-convection limit in a two-dimensional Keller-Segel-Navier-Stokes system, J. Differ. Equ., 267, 2, 938-978 (2019) · Zbl 1412.35175
[47] Wyatt, T., Pheromones and Animal Behaviour: Communication by Smell and Taste (2003), Cambridge: Cambridge University Press, Cambridge
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