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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[5]

A. Friedman, "Partial Differential Equations,", Holt, (1969). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[10]

O. A. Ladyzhenskaya, "The Mathematical Theory of Viscous Incompressible Flow,", Second edition, (1969). Google Scholar

[11]

O. A. Ladyzhenskaya and N. N. Ural'tseva, "Linear and Quasilinear Elliptic Equations,", Translated from the Russian by Scripta Technica, (1968). Google Scholar

[12]

J.-L. Lions, "Équations Différentielles Opérationnelles et Problémes aux Limites,", Die Grundlehren der mathematischen Wissenschaften, (1961). Google Scholar

[13]

A. Lorz, Coupled chemotaxis fluid model,, Math. Mod. Meth. Appl. Sci., 20 (2010), 987. doi: 10.1142/S0218202510004507. Google Scholar

[14]

L. Nirenberg, An extended interpolation inequality,, Ann. Scuola Norm. Sup. Pisa (3), 20 (1966), 733. Google Scholar

[15]

G. R. Sell and Y. You, "Dynamics of Evolutionary Equations,", Applied Mathematical Sciences, 143 (2002). Google Scholar

[16]

T. Senba and T. Suzuki, A quasi-linear parabolic system of chemotaxis,, Abstr. Appl. Analysis, 2006 (2306). Google Scholar

[17]

J. Simon, Compact sets in the space $L^p(0, T; B)$,, Ann. Math. Pura Appl. (4), 146 (1987), 65. Google Scholar

[18]

H. Sohr, "The Navier-Stokes Equations. An Elementary Functional Analytic Approach,", Birkhäuser Advanced Texts: Basler Lehrbücher, (2001). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[24]

J. L. Vázquez, "The Porous Medium Equation. Mathematical Theory,", Oxford Mathematical Monographs, (2007). Google Scholar

[25]

M. Winkler, Does a 'volume-filling effect' always prevent chemotactic collapse?,, Mathematical Methods in the Applied Sciences, 33 (2010), 12. Google Scholar

[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. Google Scholar

show all references

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[5]

A. Friedman, "Partial Differential Equations,", Holt, (1969). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[10]

O. A. Ladyzhenskaya, "The Mathematical Theory of Viscous Incompressible Flow,", Second edition, (1969). Google Scholar

[11]

O. A. Ladyzhenskaya and N. N. Ural'tseva, "Linear and Quasilinear Elliptic Equations,", Translated from the Russian by Scripta Technica, (1968). Google Scholar

[12]

J.-L. Lions, "Équations Différentielles Opérationnelles et Problémes aux Limites,", Die Grundlehren der mathematischen Wissenschaften, (1961). Google Scholar

[13]

A. Lorz, Coupled chemotaxis fluid model,, Math. Mod. Meth. Appl. Sci., 20 (2010), 987. doi: 10.1142/S0218202510004507. Google Scholar

[14]

L. Nirenberg, An extended interpolation inequality,, Ann. Scuola Norm. Sup. Pisa (3), 20 (1966), 733. Google Scholar

[15]

G. R. Sell and Y. You, "Dynamics of Evolutionary Equations,", Applied Mathematical Sciences, 143 (2002). Google Scholar

[16]

T. Senba and T. Suzuki, A quasi-linear parabolic system of chemotaxis,, Abstr. Appl. Analysis, 2006 (2306). Google Scholar

[17]

J. Simon, Compact sets in the space $L^p(0, T; B)$,, Ann. Math. Pura Appl. (4), 146 (1987), 65. Google Scholar

[18]

H. Sohr, "The Navier-Stokes Equations. An Elementary Functional Analytic Approach,", Birkhäuser Advanced Texts: Basler Lehrbücher, (2001). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[24]

J. L. Vázquez, "The Porous Medium Equation. Mathematical Theory,", Oxford Mathematical Monographs, (2007). Google Scholar

[25]

M. Winkler, Does a 'volume-filling effect' always prevent chemotactic collapse?,, Mathematical Methods in the Applied Sciences, 33 (2010), 12. Google Scholar

[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. Google Scholar

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