Abstract
In this paper, we consider the Cauchy problem for the three dimensional chemotaxis–Euler equations. By exploring the new a priori estimates, we prove the global existence of weak solutions for the 3D chemotaxis–Euler equations.
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Biler, P., Karch, G.: Blowup of solutions to generalized Keller–Segel model. J. Evol. Equ. 10, 247–262 (2010)
Blanchet, A., Dolbeault, J., Perthame, B.: Two-dimensional Keller–Segel model: optimal critical mass and qualitative properties of the solutions. Electron. J. Differ. Equ. 44, 1–33 (2006)
Bony, J.M.: Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires. Ann. Sci. École Norm. Sup. 14, 209–246 (1981)
Calvez, V., Corrias, L.: The parabolic–parabolic Keller–Segel model in \({\mathbb{R}}^2\). Commun. Math. Sci. 6, 417–447 (2008)
Chae, M., Kang, K., Lee, J.: Existence of smooth solutions to coupled chemotaxis–fluid equations. Discrete Contin. Dynam. Syst. 33, 2271–2297 (2013)
Chertock, A., Fellner, K., Kurganov, A., Lorz, A., Markowich, P.A.: Sinking, merging and stationary plumes in a coupled chemotaxis–fluid model: a high-resolution numerical approach. J. Fluid Mech. 694, 155–190 (2012)
Duan, R.J., Lorz, A., Markowich, P.A.: Global solutions to the coupled chemotaxis–fluid equations. Comm. Partial Differ. Equ. 35, 1635–1673 (2010)
Keller, E.F., Segel, L.A.: Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol. 26, 399–415 (1970)
Liu, J.G., Lorz, A.: A coupled chemotaxis-fluid model: global existence. Ann. Inst. H. Poincaré Anal. Non Linéaire 28, 643–652 (2011)
Lorz, A.: Coupled chemotaxis fluid model. Math. Models Methods Appl. Sci. 20, 987–1004 (2010)
Lorz, A.: A coupled Keller–Segel–Stokes model: global existence for small initial data and blow-up delay. Commun. Math. Sci. 10, 555–574 (2012)
Miao, C., Wu, J., Zhang, Z.: Littlewood–Paley Theory and Applications to Fluid Dynamics Equations. Science Press, Beijing (2012)
Tuval, I., Cisneros, L., Dombrowski, C., Wolgemuth, C.W., Kessler, J.O., Goldstein, R.E.: Bacterial swimming and oxygen transport near constant lines. Proc. Natl. Acad. Sci. USA 102, 2277–2282 (2005)
Winkler, M.: Global large-data solutions in a chemotaxis–(Navier–)Stokes system modeling cellular swimming in fluid drops. Comm. Partial Differ. Equ. 37, 319–352 (2012)
Winkler, M.: Stabilization in a two-dimensional chemotaxis–Navier–Stokes system. Arch. Rational Mech. Anal. 211, 455–487 (2014)
Winkler, M.: Global weak solutions in a three-dimensional chemotaxis–Navier–Stokes system. Ann. I. H. Poincaré AN (2015). doi:10.1016/j.anihpc.2015.05.002
Zhang, Q.: Local well-posedness for the chemotaxis–Navier–Stokes equations in Besov spaces. Nonlinear Anal. Real World Appl. 17, 89–100 (2014)
Zhang, Q., Zheng, X.: Global well-posedness for the two-dimensional incompressible chemptaxis–Navier–Stokes equations. SIAM J. Math. Anal. 46, 3078–3105 (2014)
Zhang, Q.: On the inviscid limit of the three dimensional incompressible chemotaxis–Navier–Stokes equations. Nonlinear Anal. Real World Appl. 27, 70–79 (2016)
Zhang, Q.: Blowup criterion of smooth solutions for the incompressible chemotaxis–Euler equations. Z. Angew. Math. Mech. 96(4), 466–476 (2016)
Acknowledgements
Q. Zhang was partially supported by the National Natural Science Foundation of China (11501160 and 61572011), the Second Batch of Young Talents of Hebei Province and Outstanding Youth Foundation of Hebei University (2015JQ01).
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Zhang, G., Zhang, Q. Global existence of weak solutions for the 3D chemotaxis–Euler equations. RACSAM 112, 195–207 (2018). https://doi.org/10.1007/s13398-016-0374-3
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DOI: https://doi.org/10.1007/s13398-016-0374-3