Existence of Local Solutions for A Chemotaxis Navier Stokes System Modeling Cellular Swimming in Fluid Drops with Logistic Source

Khayriyah Arafah; Aesha Lagha

doi:10.54172/dg3v6n81

Authors

Khayriyah Arafah Department of Mathematics, Faculty of Education, Aljafarah University, Libya Author
Aesha Lagha Department of Mathematics, Faculty of Science, Zawia University, Libya Author

DOI:

https://doi.org/10.54172/dg3v6n81

Keywords:

Chemotaxis system, Energy method, nonlinear diffusion

Abstract

In this paper, we are concerned with the Cauchy problem for the three-dimensional chemotaxis system with an indirect signal production mechanism involving a diffusive partial differential equation. Which describes the motion of bacteria, Eukaryotes, in a fluid. Precisely, for the Chemotaxis-Navier–Stokes system modeling cellular swimming in fluid drops. We established the existence of local solutions to the compressible chemotaxis equation. We proved the local existence of the Cauchy problem (1.1)-(1.2) in with the small initial data by using the energy method.

References

Bellomo, N., Bellouquid, A., Tao, Y., & Winkler, M. (2015). Toward a mathematical theory of Keller–Segel models of pattern formation in biological tissues. Mathematical Models and Methods in Applied Sciences, 25(09), 1663-1763.

Chae, M., Kang, K., & Lee, J. (2012). Existence of smooth solutions to coupled chemotaxis-fluid equations. Discrete and Continuous Dynamical Systems, 33(6), 2271-2297.

Chae, M., Kang, K., & Lee, J. (2014). Global existence and temporal decay in Keller-Segel models coupled to fluid equations. Communications in Partial Differential Equations, 39(7), 1205-1235.

Cieślak, T., & Laurençot, P. (2010). Finite time blow-up for a one-dimensional quasilinear parabolic–parabolic chemotaxis system. Annales de l'Institut Henri Poincaré C, Analyse non linéaire,

Cieślak, T., & Stinner, C. (2012). Finite-time blowup and global-in-time unbounded solutions to a parabolic–parabolic quasilinear Keller–Segel system in higher dimensions. Journal of Differential Equations, 252(10), 5832-5851.

Cieślak, T., & Winkler, M. (2008). Finite-time blow-up in a quasilinear system of chemotaxis. Nonlinearity, 21(5), 1057.

Cieślak, T., & Winkler, M. (2017). Global bounded solutions in a two-dimensional quasilinear Keller–Segel system with exponentially decaying diffusivity and subcritical sensitivity. Nonlinear Analysis: Real World Applications, 35, 1-19.

Di Francesco, M., Lorz, A., & Markowich, P. (2010). Chemotaxis-fluid coupled model for swimming bacteria with nonlinear diffusion: global existence and asymptotic behavior. Discrete and Continuous Dynamical Systems, 28(4), 1437-1453.

Dombrowski, C., Cisneros, L., Chatkaew, S., Goldstein, R. E., & Kessler, J. O. (2004). Self-concentration and large-scale coherence in bacterial dynamics. Physical review letters, 93(9), 098103.

Fujie, K., & Senba, T. (2017). Application of an Adams type inequality to a two-chemical substances chemotaxis system. Journal of Differential Equations, 263(1), 88-148.

Hattori, H., & Lagha, A. (2021a). Existence of global solutions to chemotaxis fluid system with logistic source. Electronic Journal of Qualitative Theory of Differential Equations, 2021(53), 1-27.

Hattori, H., & Lagha, A. (2021b). Global existence and decay rates of the solutions for a chemotaxis system with Lotka-Volterra type model for chemoattractant and repellent. Discrete & Continuous Dynamical Systems: Series A, 41(11).

Hillen, T., & Painter, K. J. (2009). A user’s guide to PDE models for chemotaxis. Journal of mathematical biology, 58(1), 183-217.

Horstmann, D., & Winkler, M. (2005). Boundedness vs. blow-up in a chemotaxis system. Journal of Differential Equations, 215(1), 52-107.

Hou, Q., & Wang, Z. (2019). Convergence of boundary layers for the Keller–Segel system with singular sensitivity in the half-plane. Journal de Mathématiques Pures et Appliquées, 130, 251-287.

Keller, E. F., & Segel, L. A. (1970). Initiation of slime mold aggregation viewed as an instability. Journal of theoretical biology, 26(3), 399-415.

Li, X. (2019). On a fully parabolic chemotaxis system with nonlinear signal secretion. Nonlinear Analysis: Real World Applications, 49, 24-44.

Lorz, A. (2010). Coupled chemotaxis fluid model. Mathematical Models and Methods in Applied Sciences, 20(06), 987-1004.

Pan, X., & Wang, L. (2021). Improvement of conditions for boundedness in a fully parabolic chemotaxis system with nonlinear signal production. Comptes Rendus. Mathématique, 359(2), 161-168.

Rosen, G. (1978). Steady-state distribution of bacteria chemotactic toward oxygen. Bulletin of Mathematical Biology, 40, 671-674.

Shi, S., Liu, Z., & Jin, H.-Y. (2017). Boundedness and Large Time Behavior of an Attraction-Repulsion Chemotaxis Model With Logistic Source. Kinetic & Related Models, 10(3).

Tao, Y., & Winkler, M. (2012). Boundedness in a quasilinear parabolic–parabolic Keller–Segel system with subcritical sensitivity. Journal of Differential Equations, 252(1), 692-715.

Tuval, I., Cisneros, L., Dombrowski, C., Wolgemuth, C. W., Kessler, J. O., & Goldstein, R. E. (2005). Bacterial swimming and oxygen transport near contact lines. Proceedings of the National Academy of Sciences, 102(7), 2277-2282.

Wang, Y., Winkler, M., & Xiang, Z. (2018). The small-convection limit in a two-dimensional chemotaxis-Navier–Stokes system. Mathematische Zeitschrift, 289, 71-108.

Winkler, M. (2012). Global large-data solutions in a chemotaxis-(Navier–) Stokes system modeling cellular swimming in fluid drops. Communications in Partial Differential Equations, 37(2), 319-351.

Winkler, M. (2014). Stabilization in a two-dimensional chemotaxis-Navier–Stokes system. Archive for Rational Mechanics and Analysis, 211, 455-487.

Winkler, M. (2016). Global weak solutions in a three-dimensional chemotaxis–Navier–Stokes system. Annales de l'Institut Henri Poincaré C, Analyse non linéaire,

Winkler, M. (2017a). Global existence and slow grow-up in a quasilinear Keller–Segel system with exponentially decaying diffusivity. Nonlinearity, 30(2), 735.

Winkler, M. (2017b). How far do chemotaxis-driven forces influence regularity in the Navier-Stokes system? Transactions of the American Mathematical Society, 369(5), 3067-3125.

Zhang, Q., & Li, Y. (2015). Convergence Rates of Solutions for a Two-Dimensional Chemotaxis-Navier-Stokes System. Discrete & Continuous Dynamical Systems-Series B, 20(8).