# American Institute of Mathematical Sciences

May  2012, 32(5): 1901-1914. doi: 10.3934/dcds.2012.32.1901

## Global existence and boundedness in a Keller-Segel-Stokes model with arbitrary porous medium diffusion

 1 Department of Applied Mathematics, Dong Hua University, Shanghai 200051, China 2 Institut für Mathematik, Universität Paderborn, 33098 Paderborn, Germany

Received  November 2010 Revised  October 2011 Published  January 2012

This paper deals with a boundary-value problem in two-dimensional smoothly bounded domains for the coupled chemotaxis-fluid model $$\left\{ \begin{array}{l} n_t+ u\cdot \nabla n=\Delta n^m - \nabla \cdot (n\chi(c)\nabla c)\\ c_t+ u\cdot \nabla c=\Delta c-nf(c)\\ u_t +\nabla P-\eta \Delta u+n \nabla \phi=0 \\ \nabla \cdot u=0, \end{array} \right.$$ which describes the motion of oxygen-driven swimming bacteria in an incompressible fluid. The given functions $\chi$ and $f$ are supposed to be sufficiently smooth and such that $f(0)=0$.
It is proved that global bounded weak solutions exist whenever $m>1$ and the initial data $(n_0,c_0,u_0)$ are sufficiently regular satisfying $n_0 \ge 0$ and $c_0\ge 0$. This extends a recent result by Di Francesco, Lorz and Markowich (Discrete Cont. Dyn. Syst. A 28 (2010)) which asserts global existence of weak solutions under the constraint $m \in (\frac{3}{2},2]$.
Citation: Youshan Tao, Michael Winkler. Global existence and boundedness in a Keller-Segel-Stokes model with arbitrary porous medium diffusion. Discrete and Continuous Dynamical Systems, 2012, 32 (5) : 1901-1914. doi: 10.3934/dcds.2012.32.1901
##### References:
 [1] N. D. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equations, Comm. Partial Differential Equations, 4 (1979), 827-868. doi: 10.1080/03605307908820113. [2] V. Calvez and J. A. Carrillo, Volume effects in the Keller-Segel model: Energy estimates preventing blow-up, J. Math. Pures Appl. (9), 86 (2006), 155-175. doi: 10.1016/j.matpur.2006.04.002. [3] M. Di Francesco, A. Lorz and P. Markowich, Chemotaxis-fluid coupled model for swimming bacteria with nonlinear diffusion: Global existence and asymptotic behavior, Discrete Cont. Dyn. Syst., 28 (2010), 1437-1453. [4] R. J. Duan, A. Lorz and P. Markowich, Global solutions to the coupled chemotaxis-fluid equations, Comm. Partial Differential Equations, 35 (2010), 1635-1673. doi: 10.1080/03605302.2010.497199. [5] A. Friedman, "Partial Differential Equations," Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1969. [6] D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Diff. Eqns., 215 (2005), 52-107. doi: 10.1016/j.jde.2004.10.022. [7] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instaility, J. Theor. Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5. [8] R. Kowalczyk, Preventing blow-up in a chemotaxis model, J. Math. Anal. Appl., 305 (2005), 566-585. doi: 10.1016/j.jmaa.2004.12.009. [9] R. Kowalczyk and Z. Szymańska, On the global existence of solutions to an aggregation model, J. Math. Anal. Appl., 343 (2008), 379-398. doi: 10.1016/j.jmaa.2008.01.005. [10] O. A. Ladyzhenskaya, "The Mathematical Theory of Viscous Incompressible Flow," Second edition, revised and enlarges, Translated from the Russian by Richard A. Silverman and John Chu, Mathematics and its Applications, Vol. 2, Gordon and Breach, Science Publishers, New York-London-Paris, 1969. [11] O. A. Ladyzhenskaya and N. N. Ural'tseva, "Linear and Quasilinear Elliptic Equations," Translated from the Russian by Scripta Technica, Inc., Translation editor: Leon Ehrenpreis, Academic Press, New York-London, 1968. [12] J.-L. Lions, "Équations Différentielles Opérationnelles et Problémes aux Limites," Die Grundlehren der mathematischen Wissenschaften, Bd. 111, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1961. [13] A. Lorz, Coupled chemotaxis fluid model, Math. Mod. Meth. Appl. Sci., 20 (2010), 987-1004. doi: 10.1142/S0218202510004507. [14] L. Nirenberg, An extended interpolation inequality, Ann. Scuola Norm. Sup. Pisa (3), 20 (1966), 733-737. [15] G. R. Sell and Y. You, "Dynamics of Evolutionary Equations," Applied Mathematical Sciences, 143, Springer-Verlag, New York, 2002. [16] T. Senba and T. Suzuki, A quasi-linear parabolic system of chemotaxis, Abstr. Appl. Analysis, 2006, Art. ID 23061, 21 pp. [17] J. Simon, Compact sets in the space $L^p(0, T; B)$, Ann. Math. Pura Appl. (4), 146 (1987), 65-96. [18] H. Sohr, "The Navier-Stokes Equations. An Elementary Functional Analytic Approach," Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel, 2001. [19] Y. Sugiyama, On $\epsilon$-regularity theorem and asymptotic behaviors of solutions for Keller-Segel systems, SIAM J. Math. Anal., 41 (2009), 1664-1692. doi: 10.1137/080721078. [20] Z. Szymańska, C. Morales Rodrigo, M. Lachowicz and M. A. Chaplain, Mathematical modelling of cancer invasion of tissue: The role and effect of nonlocal interactions, Math. Mod. Meth. Appl. Sci., 19 (2009), 257-281. [21] Y. Tao and M. Winkler, A chemotaxis-haptotaxis model: The roles of nonlinear diffusion and logistic source, SIAM J. Math. Anal., 43 (2011), 685-704. doi: 10.1137/100802943. [22] Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692-715. doi: 10.1016/j.jde.2011.08.019. [23] I. Tuval, L. Cisnerous, C. Dombrowski, C. W. Wolgemuth, J. O. Kessler and R. E. Goldstein, Bacterial swimming and oxygen transport near contact lines, Proc. Natl. Acad. Sci. USA, 102 (2005), 2277-2282. doi: 10.1073/pnas.0406724102. [24] J. L. Vázquez, "The Porous Medium Equation. Mathematical Theory," Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2007. [25] M. Winkler, Does a 'volume-filling effect' always prevent chemotactic collapse?, Mathematical Methods in the Applied Sciences, 33 (2010), 12-24. [26] M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2905. doi: 10.1016/j.jde.2010.02.008.

show all references

##### References:
 [1] N. D. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equations, Comm. Partial Differential Equations, 4 (1979), 827-868. doi: 10.1080/03605307908820113. [2] V. Calvez and J. A. Carrillo, Volume effects in the Keller-Segel model: Energy estimates preventing blow-up, J. Math. Pures Appl. (9), 86 (2006), 155-175. doi: 10.1016/j.matpur.2006.04.002. [3] M. Di Francesco, A. Lorz and P. Markowich, Chemotaxis-fluid coupled model for swimming bacteria with nonlinear diffusion: Global existence and asymptotic behavior, Discrete Cont. Dyn. Syst., 28 (2010), 1437-1453. [4] R. J. Duan, A. Lorz and P. Markowich, Global solutions to the coupled chemotaxis-fluid equations, Comm. Partial Differential Equations, 35 (2010), 1635-1673. doi: 10.1080/03605302.2010.497199. [5] A. Friedman, "Partial Differential Equations," Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1969. [6] D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Diff. Eqns., 215 (2005), 52-107. doi: 10.1016/j.jde.2004.10.022. [7] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instaility, J. Theor. Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5. [8] R. Kowalczyk, Preventing blow-up in a chemotaxis model, J. Math. Anal. Appl., 305 (2005), 566-585. doi: 10.1016/j.jmaa.2004.12.009. [9] R. Kowalczyk and Z. Szymańska, On the global existence of solutions to an aggregation model, J. Math. Anal. Appl., 343 (2008), 379-398. doi: 10.1016/j.jmaa.2008.01.005. [10] O. A. Ladyzhenskaya, "The Mathematical Theory of Viscous Incompressible Flow," Second edition, revised and enlarges, Translated from the Russian by Richard A. Silverman and John Chu, Mathematics and its Applications, Vol. 2, Gordon and Breach, Science Publishers, New York-London-Paris, 1969. [11] O. A. Ladyzhenskaya and N. N. Ural'tseva, "Linear and Quasilinear Elliptic Equations," Translated from the Russian by Scripta Technica, Inc., Translation editor: Leon Ehrenpreis, Academic Press, New York-London, 1968. [12] J.-L. Lions, "Équations Différentielles Opérationnelles et Problémes aux Limites," Die Grundlehren der mathematischen Wissenschaften, Bd. 111, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1961. [13] A. Lorz, Coupled chemotaxis fluid model, Math. Mod. Meth. Appl. Sci., 20 (2010), 987-1004. doi: 10.1142/S0218202510004507. [14] L. Nirenberg, An extended interpolation inequality, Ann. Scuola Norm. Sup. Pisa (3), 20 (1966), 733-737. [15] G. R. Sell and Y. You, "Dynamics of Evolutionary Equations," Applied Mathematical Sciences, 143, Springer-Verlag, New York, 2002. [16] T. Senba and T. Suzuki, A quasi-linear parabolic system of chemotaxis, Abstr. Appl. Analysis, 2006, Art. ID 23061, 21 pp. [17] J. Simon, Compact sets in the space $L^p(0, T; B)$, Ann. Math. Pura Appl. (4), 146 (1987), 65-96. [18] H. Sohr, "The Navier-Stokes Equations. An Elementary Functional Analytic Approach," Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel, 2001. [19] Y. Sugiyama, On $\epsilon$-regularity theorem and asymptotic behaviors of solutions for Keller-Segel systems, SIAM J. Math. Anal., 41 (2009), 1664-1692. doi: 10.1137/080721078. [20] Z. Szymańska, C. Morales Rodrigo, M. Lachowicz and M. A. Chaplain, Mathematical modelling of cancer invasion of tissue: The role and effect of nonlocal interactions, Math. Mod. Meth. Appl. Sci., 19 (2009), 257-281. [21] Y. Tao and M. Winkler, A chemotaxis-haptotaxis model: The roles of nonlinear diffusion and logistic source, SIAM J. Math. Anal., 43 (2011), 685-704. doi: 10.1137/100802943. [22] Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692-715. doi: 10.1016/j.jde.2011.08.019. [23] I. Tuval, L. Cisnerous, C. Dombrowski, C. W. Wolgemuth, J. O. Kessler and R. E. Goldstein, Bacterial swimming and oxygen transport near contact lines, Proc. Natl. Acad. Sci. USA, 102 (2005), 2277-2282. doi: 10.1073/pnas.0406724102. [24] J. L. Vázquez, "The Porous Medium Equation. Mathematical Theory," Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2007. [25] M. Winkler, Does a 'volume-filling effect' always prevent chemotactic collapse?, Mathematical Methods in the Applied Sciences, 33 (2010), 12-24. [26] M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2905. doi: 10.1016/j.jde.2010.02.008.
 [1] María Astudillo, Marcelo M. Cavalcanti. On the upper semicontinuity of the global attractor for a porous medium type problem with large diffusion. Evolution Equations and Control Theory, 2017, 6 (1) : 1-13. doi: 10.3934/eect.2017001 [2] Sachiko Ishida. Global existence and boundedness for chemotaxis-Navier-Stokes systems with position-dependent sensitivity in 2D bounded domains. Discrete and Continuous Dynamical Systems, 2015, 35 (8) : 3463-3482. doi: 10.3934/dcds.2015.35.3463 [3] Marcel Freitag. Global existence and boundedness in a chemorepulsion system with superlinear diffusion. Discrete and Continuous Dynamical Systems, 2018, 38 (11) : 5943-5961. doi: 10.3934/dcds.2018258 [4] Jiapeng Huang, Chunhua Jin. Time periodic solution to a coupled chemotaxis-fluid model with porous medium diffusion. Discrete and Continuous Dynamical Systems, 2020, 40 (9) : 5415-5439. doi: 10.3934/dcds.2020233 [5] Johannes Lankeit, Yulan Wang. Global existence, boundedness and stabilization in a high-dimensional chemotaxis system with consumption. Discrete and Continuous Dynamical Systems, 2017, 37 (12) : 6099-6121. doi: 10.3934/dcds.2017262 [6] Chunhua Jin. Boundedness and global solvability to a chemotaxis-haptotaxis model with slow and fast diffusion. Discrete and Continuous Dynamical Systems - B, 2018, 23 (4) : 1675-1688. doi: 10.3934/dcdsb.2018069 [7] Laiqing Meng, Jia Yuan, Xiaoxin Zheng. Global existence of almost energy solution to the two-dimensional chemotaxis-Navier-Stokes equations with partial diffusion. Discrete and Continuous Dynamical Systems, 2019, 39 (6) : 3413-3441. doi: 10.3934/dcds.2019141 [8] Xiaoyu Chen, Jijie Zhao, Qian Zhang. Global existence of weak solutions for the 3D axisymmetric chemotaxis-Navier-Stokes equations with nonlinear diffusion. Discrete and Continuous Dynamical Systems, 2022, 42 (9) : 4489-4522. doi: 10.3934/dcds.2022062 [9] Luis Caffarelli, Juan-Luis Vázquez. Asymptotic behaviour of a porous medium equation with fractional diffusion. Discrete and Continuous Dynamical Systems, 2011, 29 (4) : 1393-1404. doi: 10.3934/dcds.2011.29.1393 [10] Ling Liu, Jiashan Zheng. Global existence and boundedness of solution of a parabolic-parabolic-ODE chemotaxis-haptotaxis model with (generalized) logistic source. Discrete and Continuous Dynamical Systems - B, 2019, 24 (7) : 3357-3377. doi: 10.3934/dcdsb.2018324 [11] Pan Zheng. Global boundedness and decay for a multi-dimensional chemotaxis-haptotaxis system with nonlinear diffusion. Discrete and Continuous Dynamical Systems - B, 2016, 21 (6) : 2039-2056. doi: 10.3934/dcdsb.2016035 [12] Matteo Bonforte, Yannick Sire, Juan Luis Vázquez. Existence, uniqueness and asymptotic behaviour for fractional porous medium equations on bounded domains. Discrete and Continuous Dynamical Systems, 2015, 35 (12) : 5725-5767. doi: 10.3934/dcds.2015.35.5725 [13] Kristian Moring, Christoph Scheven, Sebastian Schwarzacher, Thomas Singer. Global higher integrability of weak solutions of porous medium systems. Communications on Pure and Applied Analysis, 2020, 19 (3) : 1697-1745. doi: 10.3934/cpaa.2020069 [14] Masaki Kurokiba, Toshitaka Nagai, T. Ogawa. The uniform boundedness and threshold for the global existence of the radial solution to a drift-diffusion system. Communications on Pure and Applied Analysis, 2006, 5 (1) : 97-106. doi: 10.3934/cpaa.2006.5.97 [15] Jiayi Han, Changchun Liu. Global existence for a two-species chemotaxis-Navier-Stokes system with $p$-Laplacian. Electronic Research Archive, 2021, 29 (5) : 3509-3533. doi: 10.3934/era.2021050 [16] Edoardo Mainini. On the signed porous medium flow. Networks and Heterogeneous Media, 2012, 7 (3) : 525-541. doi: 10.3934/nhm.2012.7.525 [17] Pan Zheng, Chunlai Mu, Xiaojun Song. On the boundedness and decay of solutions for a chemotaxis-haptotaxis system with nonlinear diffusion. Discrete and Continuous Dynamical Systems, 2016, 36 (3) : 1737-1757. doi: 10.3934/dcds.2016.36.1737 [18] Mengyao Ding, Wei Wang. Global boundedness in a quasilinear fully parabolic chemotaxis system with indirect signal production. Discrete and Continuous Dynamical Systems - B, 2019, 24 (9) : 4665-4684. doi: 10.3934/dcdsb.2018328 [19] Wei Wang, Yan Li, Hao Yu. Global boundedness in higher dimensions for a fully parabolic chemotaxis system with singular sensitivity. Discrete and Continuous Dynamical Systems - B, 2017, 22 (10) : 3663-3669. doi: 10.3934/dcdsb.2017147 [20] Hua Zhong, Chunlai Mu, Ke Lin. Global weak solution and boundedness in a three-dimensional competing chemotaxis. Discrete and Continuous Dynamical Systems, 2018, 38 (8) : 3875-3898. doi: 10.3934/dcds.2018168

2021 Impact Factor: 1.588