Zhai, Xiaoping; Chen, Yiren Global solutions and large time behavior for the chemotaxis-shallow water system. (English) Zbl 1456.35170 J. Differ. Equations 275, 332-358 (2021). The authors study a chemotaxis system coupled with the viscous shallow water equations which models bacteriae chemotactic movement towards higher concentration of oxygen they consume, under gravitational potential and convective transport effects. A well-posedness result is proved for that two-dimensional evolution problem with small initial data in the critical Besov spaces. Time decay of solutions – coinciding with that for the diffusion asymptotics – is proved. Reviewer: Piotr Biler (Wrocław) Cited in 2 Documents MSC: 35Q35 PDEs in connection with fluid mechanics 35Q92 PDEs in connection with biology, chemistry and other natural sciences 76D99 Incompressible viscous fluids 35B40 Asymptotic behavior of solutions to PDEs 92C17 Cell movement (chemotaxis, etc.) Keywords:chemotaxis; shallow water equations; global-in-time solutions; large time asymptotics PDFBibTeX XMLCite \textit{X. Zhai} and \textit{Y. Chen}, J. Differ. Equations 275, 332--358 (2021; Zbl 1456.35170) Full Text: DOI References: [1] Bahouri, H.; Chemin, J. Y.; Danchin, R., Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren Math. Wiss., vol. 343 (2011), Springer: Springer Berlin [2] Bresch, D.; Desjardins, B., Existence of global weak solutions for 2D viscous shallow water equations and convergence to the quasi-geostrophic model, Commun. Math. Phys., 238, 211-233 (2003) · Zbl 1037.76012 [3] Bresch, D.; Desjardins, B.; Metivier, G., Recent mathematical results and open problem about shallow water equations, (Analysis and Simulation of Fluid Dynamics, Advances in Mathematical Fluid Mechanics (2006), Birkhäuser Verlag Basel: Birkhäuser Verlag Basel Switzerland), 15-31 · Zbl 1291.35001 [4] Bui, A., Existence and uniqueness of a classical solution of an initial boundary value problem of the theory of shallow waters, SIAM J. Math. Anal., 12, 229-241 (1981) · Zbl 0468.76021 [5] Chae, M.; Kang, K.; Lee, J., Global existence and temporal decay in Keller-Segel models coupled to fluid equations, Commun. Partial Differ. Equ., 39, 1205-1235 (2014) · Zbl 1304.35481 [6] Che, J.; Chen, L.; Duan, B.; Luo, Z., On the existence of local strong solutions to chemotaxis-shallow water system with large data and vacuum, J. Differ. Equ., 261, 6758-6789 (2016) · Zbl 1351.35128 [7] Chen, Q.; Miao, C.; Zhang, Z., On the well-posedness for the viscous shallow water equations, SIAM J. Math. Anal., 40, 443-474 (2008) · Zbl 1169.35048 [8] Chen, Q.; Miao, C.; Zhang, Z., Well-posedness in critical spaces for the compressible Navier-Stokes equations with density dependent viscosities, Rev. Mat. Iberoam., 26, 915-946 (2010) · Zbl 1205.35189 [9] Danchin, R., Well-posedness in critical spaces for barotropic viscous fluids with truly not constant density, Commun. Partial Differ. Equ., 32, 1373-1397 (2007) · Zbl 1120.76052 [10] Danchin, R.; Mucha, P., Compressible Navier-Stokes system: large solutions and incompressible limit, Adv. Math., 320, 904-925 (2017) · Zbl 1384.35058 [11] Duan, B.; Luo, Z.; Zheng, Y., Local existence of classical solutions to shallow water equations with Cauchy data containing vacuum, SIAM J. Math. Anal., 44, 541-567 (2012) · Zbl 1387.35487 [12] Duan, Q., On the dynamics of Navier-Stokes equations for a shallow water model, J. Differ. Equ., 250, 2687-2714 (2011) · Zbl 1215.35172 [13] Duan, R.; Li, X.; Xiang, Z., Global existence and large time behavior for a two-dimensional chemotaxis-Navier-Stokes system, J. Differ. Equ., 263, 6284-6316 (2017) · Zbl 1378.35160 [14] Duan, R.; Lorz, A.; Markowich, P., Global solutions to the coupled chemotaxis-fluid equations, Commun. Partial Differ. Equ., 35, 1635-1673 (2010) · Zbl 1275.35005 [15] Duan, R.; Xiang, Z., A note on global existence for the chemotaxis-Stokes model with nonlinear diffusion, Int. Math. Res. Not., 2012, Article rns270 pp. (2012) [16] Guo, Y.; Wang, Y. J., Decay of dissipative equations and negative Sobolev spaces, Commun. Partial Differ. Equ., 37, 2165-2208 (2012) · Zbl 1258.35157 [17] Kloeden, P., Global existence of classic solution in the dissipative shallow water equations, SIAM J. Math. Anal., 16, 301-315 (1985) · Zbl 0579.76047 [18] Lorz, A., Coupled chemotaxis fluid model, Math. Models Methods Appl. Sci., 20, 987-1004 (2010) · Zbl 1191.92004 [19] Sondbye, L., Global existence for Dirichlet problem for the viscous shallow water equations, J. Math. Anal. Appl., 202, 236-258 (1996) [20] Sondbye, L., Global existence for the Cauchy problem for the viscous shallow water equations, Rocky Mt. J. Math., 28, 1135-1152 (1998) · Zbl 0928.35129 [21] Souplet, P.; Winkler, M., Blow-up profiles for the parabolic-elliptic Keller-Segel system in dimensions \(n \geq 3\), Commun. Math. Phys., 367, 665-681 (2019) · Zbl 1411.35140 [22] Tao, Q.; Yao, Z., Global existence and large time behavior for a two-dimensional chemotaxis-shallow water system, J. Differ. Equ., 265, 3092-3129 (2018) · Zbl 1402.35287 [23] Tuval, I.; Cisneros, L.; Dombrowski, C.; Wolgemuth, C.; Kessler, J.; Glodstein, R., Bacterial swimming and oxygen transport near contact lines, Proc. Natl. Acad. Sci. USA, 102, 2277-2282 (2005) · Zbl 1277.35332 [24] Velázquez, J., Point dynamics in a singular limit of the Keller-Segel model 1: motion of the concentration regions, SIAM J. Appl. Math., 64, 1198-1223 (2004) · Zbl 1058.35021 [25] Wang, J.; Chen, L.; Liang, H., Parabolic elliptic type Keller-Segel system on the whole space case, Dyn. Syst., 36, 1061-1084 (2016) · Zbl 1326.35408 [26] Wang, W.; Wang, Y., The \(L^p\) decay estimates for the chemotaxis-shallow water system, J. Math. Anal. Appl., 474, 640-665 (2019) · Zbl 1416.35219 [27] Wang, W.; Xu, C., The Cauchy problem for viscous shallow water equations, Rev. Mat. Iberoam., 21, 1-24 (2005) · Zbl 1095.35037 [28] Wang, Y.; Winkler, M.; Xiang, Z., The fast signal diffusion limit in Keller-Segel(-fluid) systems, Calc. Var. Partial Differ. Equ., 58, Article 196 pp. (2019) · Zbl 1426.92009 [29] Winkler, M., Global large-data solutions in a chemotaxis-(Navier-)Stokes system modeling cellular swimming in fluid drops, Commun. Partial Differ. Equ., 37, 319-351 (2012) · Zbl 1236.35192 [30] Winkler, M., Stabilization in a two-dimensional chemotaxis-Navier-Stokes system, Arch. Ration. Mech. Anal., 211, 455-487 (2014) · Zbl 1293.35220 [31] 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, 3789-3828 (2015) · Zbl 1333.35104 [32] Winkler, M., Global weak solutions in a three-dimensional chemotaxis-Navier-Stokes system, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 33, 1329-1352 (2016) · Zbl 1351.35239 [33] Winkler, M., How far do chemotaxis-driven forces influence regularity in the Navier-Stokes system?, Trans. Am. Math. Soc., 369, 3067-3125 (2017) · Zbl 1356.35071 [34] Winkler, M., A three-dimensional Keller-Segel-Navier-Stokes system with logistic source: global weak solutions and asymptotic stabilization, J. Funct. Anal., 276, 1339-1401 (2019) · Zbl 1408.35132 [35] Winkler, M., Global classical solvability and generic infinite-time blow-up in quasilinear Keller-Segel systems with bounded sensitivities, J. Differ. Equ., 266, 8034-8066 (2019) · Zbl 1415.35052 [36] Xin, Z.; Xu, J., Optimal decay for the compressible Navier-Stokes equations without additional smallness assumptions [37] Zhai, X.; Li, Y., Global well-posedness and large time behavior of solutions to the n-dimensional compressible Oldroyd-B model [38] Zhai, X.; Ye, H., On global large energy solutions to the viscous shallow water equations, Discrete Contin. Dyn. Syst., Ser. B, 25, 4277-4293 (2020) · Zbl 1451.35145 [39] Zhai, X.; Yin, Z., Global solutions to the chemotaxis-Navier-Stokes equations with some large initial data, Discrete Contin. Dyn. Syst., Ser. A, 37, 2829-2859 (2017) · Zbl 1358.35214 This reference list is based on information provided by the publisher or from digital mathematics libraries. 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