# 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 & Continuous Dynamical Systems - A, 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. 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. 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. [4] R. J. Duan, A. Lorz and P. Markowich, Global solutions to the coupled chemotaxis-fluid equations,, Comm. Partial Differential Equations, 35 (2010), 1635. doi: 10.1080/03605302.2010.497199. [5] A. Friedman, "Partial Differential Equations,", Holt, (1969). [6] D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system,, J. Diff. Eqns., 215 (2005), 52. 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. 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. 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. doi: 10.1016/j.jmaa.2008.01.005. [10] O. A. Ladyzhenskaya, "The Mathematical Theory of Viscous Incompressible Flow,", Second edition, (1969). [11] O. A. Ladyzhenskaya and N. N. Ural'tseva, "Linear and Quasilinear Elliptic Equations,", Translated from the Russian by Scripta Technica, (1968). [12] J.-L. Lions, "Équations Différentielles Opérationnelles et Problémes aux Limites,", Die Grundlehren der mathematischen Wissenschaften, (1961). [13] A. Lorz, Coupled chemotaxis fluid model,, Math. Mod. Meth. Appl. Sci., 20 (2010), 987. doi: 10.1142/S0218202510004507. [14] L. Nirenberg, An extended interpolation inequality,, Ann. Scuola Norm. Sup. Pisa (3), 20 (1966), 733. [15] G. R. Sell and Y. You, "Dynamics of Evolutionary Equations,", Applied Mathematical Sciences, 143 (2002). [16] T. Senba and T. Suzuki, A quasi-linear parabolic system of chemotaxis,, Abstr. Appl. Analysis, 2006 (2306). [17] J. Simon, Compact sets in the space $L^p(0, T; B)$,, Ann. Math. Pura Appl. (4), 146 (1987), 65. [18] H. Sohr, "The Navier-Stokes Equations. An Elementary Functional Analytic Approach,", Birkhäuser Advanced Texts: Basler Lehrbücher, (2001). [19] Y. Sugiyama, On $\epsilon$-regularity theorem and asymptotic behaviors of solutions for Keller-Segel systems,, SIAM J. Math. Anal., 41 (2009), 1664. 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. [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. 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. 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. doi: 10.1073/pnas.0406724102. [24] J. L. Vázquez, "The Porous Medium Equation. Mathematical Theory,", Oxford Mathematical Monographs, (2007). [25] M. Winkler, Does a 'volume-filling effect' always prevent chemotactic collapse?,, Mathematical Methods in the Applied Sciences, 33 (2010), 12. [26] M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model,, J. Differential Equations, 248 (2010), 2889. doi: 10.1016/j.jde.2010.02.008.

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##### References:
 [1] N. D. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equations,, Comm. Partial Differential Equations, 4 (1979), 827. 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. 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. [4] R. J. Duan, A. Lorz and P. Markowich, Global solutions to the coupled chemotaxis-fluid equations,, Comm. Partial Differential Equations, 35 (2010), 1635. doi: 10.1080/03605302.2010.497199. [5] A. Friedman, "Partial Differential Equations,", Holt, (1969). [6] D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system,, J. Diff. Eqns., 215 (2005), 52. 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. 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. 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. doi: 10.1016/j.jmaa.2008.01.005. [10] O. A. Ladyzhenskaya, "The Mathematical Theory of Viscous Incompressible Flow,", Second edition, (1969). [11] O. A. Ladyzhenskaya and N. N. Ural'tseva, "Linear and Quasilinear Elliptic Equations,", Translated from the Russian by Scripta Technica, (1968). [12] J.-L. Lions, "Équations Différentielles Opérationnelles et Problémes aux Limites,", Die Grundlehren der mathematischen Wissenschaften, (1961). [13] A. Lorz, Coupled chemotaxis fluid model,, Math. Mod. Meth. Appl. Sci., 20 (2010), 987. doi: 10.1142/S0218202510004507. [14] L. Nirenberg, An extended interpolation inequality,, Ann. Scuola Norm. Sup. Pisa (3), 20 (1966), 733. [15] G. R. Sell and Y. You, "Dynamics of Evolutionary Equations,", Applied Mathematical Sciences, 143 (2002). [16] T. Senba and T. Suzuki, A quasi-linear parabolic system of chemotaxis,, Abstr. Appl. Analysis, 2006 (2306). [17] J. Simon, Compact sets in the space $L^p(0, T; B)$,, Ann. Math. Pura Appl. (4), 146 (1987), 65. [18] H. Sohr, "The Navier-Stokes Equations. An Elementary Functional Analytic Approach,", Birkhäuser Advanced Texts: Basler Lehrbücher, (2001). [19] Y. Sugiyama, On $\epsilon$-regularity theorem and asymptotic behaviors of solutions for Keller-Segel systems,, SIAM J. Math. Anal., 41 (2009), 1664. 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. [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. 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. 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. doi: 10.1073/pnas.0406724102. [24] J. L. Vázquez, "The Porous Medium Equation. Mathematical Theory,", Oxford Mathematical Monographs, (2007). [25] M. Winkler, Does a 'volume-filling effect' always prevent chemotactic collapse?,, Mathematical Methods in the Applied Sciences, 33 (2010), 12. [26] M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model,, J. Differential Equations, 248 (2010), 2889. doi: 10.1016/j.jde.2010.02.008.
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