August  2015, 35(8): 3463-3482. doi: 10.3934/dcds.2015.35.3463

Global existence and boundedness for chemotaxis-Navier-Stokes systems with position-dependent sensitivity in 2D bounded domains

1. 

Department of Mathematics, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan

Received  August 2014 Revised  December 2014 Published  February 2015

This paper is concerned with degenerate chemotaxis-Navier-Stokes systems with position-dependent sensitivity on a two dimensional bounded domain. It is known that in the case without a position-dependent sensitivity function, Tao-Winkler (2012) constructed a globally bounded weak solution of a chemotaxis-Stokes system with any porous medium diffusion, and Winkler (2012, 2014) succeeded in proving global existence and stabilization of classical solutions to a chemotaxis-Navier-Stokes system with linear diffusion. The present work shows global existence and boundedness of weak solutions to a chemotaxis-Navier-Stokes system with position-dependent sensitivity for any porous medium diffusion.
Citation: Sachiko Ishida. Global existence and boundedness for chemotaxis-Navier-Stokes systems with position-dependent sensitivity in 2D bounded domains. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3463-3482. doi: 10.3934/dcds.2015.35.3463
References:
[1]

X. Cao and S. Ishida, Global-in-time bounded weak solutions to a degenerate quasilinear Keller-Segel system with rotation,, Nonlinearity, 27 (2014), 1899.  doi: 10.1088/0951-7715/27/8/1899.  Google Scholar

[2]

M. Chae, K. Kang, J. Lee and Jihoon, Global existence and temporal decay in Keller-Segel models coupled to fluid equations,, Comm. Partial Differential Equations, 39 (2014), 1205.  doi: 10.1080/03605302.2013.852224.  Google Scholar

[3]

A. Chertock, K. Fellner, A. Kurganov, A. Lorz and P. A. Markowich, Sinking, merging and stationary plumes in a coupled chemotaxis-fluid model: a high-resolution numerical approach,, J. Fluid Mech., 694 (2012), 155.  doi: 10.1017/jfm.2011.534.  Google Scholar

[4]

R. 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]

R. Duan and Z. Xiang, A note on global existence for the chemotaxis-Stokes model with nonlinear diffusion,, Int. Math. Res. Not. IMRN, (2014), 1833.   Google Scholar

[6]

M. Di Francesco, A. Lorz and P. Markowich, Chemotaxis-fluid coupled model for swimming bacteria with nonlinear diffusion: global existence and asymptotic behavior,, Discrete Contin. Dyn. Syst., 28 (2010), 1437.  doi: 10.3934/dcds.2010.28.1437.  Google Scholar

[7]

M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 24 (1997), 633.   Google Scholar

[8]

T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis,, J. Math. Biol., 58 (2009), 183.  doi: 10.1007/s00285-008-0201-3.  Google Scholar

[9]

S. Ishida, K. Seki and T. Yokota, Boundedness in quasilinear Keller-Segel systems of parabolic-parabolic type on non-convex bounded domains,, J. Differential Equations, 256 (2014), 2993.  doi: 10.1016/j.jde.2014.01.028.  Google Scholar

[10]

S. Ishida and T. Yokota, Global existence of weak solutions to quasilinear degenerate Keller-Segel systems of parabolic-parabolic type,, J. Differential Equations, 252 (2012), 1421.  doi: 10.1016/j.jde.2011.02.012.  Google Scholar

[11]

S. Ishida and T. Yokota, Blow-up in finite or infinite time for quasilinear degenerate Keller-Segel systems of parabolic-parabolic type,, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2569.  doi: 10.3934/dcdsb.2013.18.2569.  Google Scholar

[12]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability,, J. Theor. Biol., 26 (1970), 399.  doi: 10.1016/0022-5193(70)90092-5.  Google Scholar

[13]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type,, American Mathematical Society, (1968).   Google Scholar

[14]

T. Li, A. Suen, C. Xue and M. Winkler, Global small-data solutions of a two-dimensional chemotaxis system with rotational flux term,, Math. Models Methods Appl. Sci., 25 (2015), 721.  doi: 10.1142/S0218202515500177.  Google Scholar

[15]

J.-G. Liu and A. Lorz, A coupled chemotaxis-fluid model: Global existence,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 643.  doi: 10.1016/j.anihpc.2011.04.005.  Google Scholar

[16]

J. López-Gómez, T. cc and T. Yamada, Non-trivial $\omega$-limit sets and oscillating solutions in a chemotaxis model in $\mathbbR^2$ with critical mass,, J. Funct. Anal., 266 (2014), 3455.  doi: 10.1016/j.jfa.2014.01.015.  Google Scholar

[17]

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

[18]

A. Lorz, A coupled Keller-Segel-Stokes model: Global existence for small initial data and blow-up delay,, Commun. Math. Sci., 10 (2012), 555.  doi: 10.4310/CMS.2012.v10.n2.a7.  Google Scholar

[19]

T. Nagai, T. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis,, Funkcial. Ekvac., 40 (1997), 411.   Google Scholar

[20]

K. Osaki and A. Yagi, Finite dimensional attractor for one-dimensional Keller-Segel equations,, Funkcial. Ekvac., 44 (2001), 441.   Google Scholar

[21]

Y. Seki, Y. Sugiyama and J. J. L. Velázquez, Multiple peak aggregations for the Keller-Segel system,, Nonlinearity, 26 (2013), 319.  doi: 10.1088/0951-7715/26/2/319.  Google Scholar

[22]

Y. Shibata and S. Shimizu, On the $L^p$-$L^q$ maximal regularity of the Neumann problem for the Stokes equations in a bounded domain,, J. Reine Angew. Math., 615 (2008), 157.  doi: 10.1515/CRELLE.2008.013.  Google Scholar

[23]

H. Sohr, The Navier-Stokes Equations: An Elementary Functional Analytic Approach,, Birkhäuser-Verlag, (2001).   Google Scholar

[24]

Y. Sugiyama, Global existence in sub-critical cases and finite time blow-up in super-critical cases to degenerate Keller-Segel systems,, Differential Integral Equations, 19 (2006), 841.   Google Scholar

[25]

Y. Sugiyama and Y. Yahagi, Extinction, decay and blow-up for Keller-Segel systems of fast diffusion type,, J. Differential Equations, 250 (2011), 3047.  doi: 10.1016/j.jde.2011.01.016.  Google Scholar

[26]

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

[27]

Y. Tao and M. Winkler, Global existence and boundedness in a Keller-Segel-Stokes model with arbitrary porous medium diffusion,, Discrete Contin. Dyn. Syst., 32 (2012), 1901.  doi: 10.3934/dcds.2012.32.1901.  Google Scholar

[28]

Y. Tao and M. Winkler, Locally bounded global solutions in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 157.  doi: 10.1016/j.anihpc.2012.07.002.  Google Scholar

[29]

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis,, North-Holland Publishing Co., (1977).   Google Scholar

[30]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Second edition,, Applied Mathematical Sciences, (1997).  doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[31]

I. Tuval, L. Cisneros, 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

[32]

M. Winkler, Global large-data solutions in a chemotaxis-(Navier-)Stokes system modeling cellular swimming in fluid drops,, Comm. Partial Differential Equations, 37 (2012), 319.  doi: 10.1080/03605302.2011.591865.  Google Scholar

[33]

M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system,, J. Math. Pures Appl. 100 (2013), 100 (2013), 748.  doi: 10.1016/j.matpur.2013.01.020.  Google Scholar

[34]

M. Winkler, Stabilization in a two-dimensional chemotaxis-Navier-Stokes system,, Arch. Ration. Mech. Anal. 211 (2014), 211 (2014), 455.  doi: 10.1007/s00205-013-0678-9.  Google Scholar

[35]

C. Xue and H. G. Othmer, Multiscale models of taxis-driven patterning in bacterial population,, SIAM J. Appl. Math., 70 (2009), 133.  doi: 10.1137/070711505.  Google Scholar

show all references

References:
[1]

X. Cao and S. Ishida, Global-in-time bounded weak solutions to a degenerate quasilinear Keller-Segel system with rotation,, Nonlinearity, 27 (2014), 1899.  doi: 10.1088/0951-7715/27/8/1899.  Google Scholar

[2]

M. Chae, K. Kang, J. Lee and Jihoon, Global existence and temporal decay in Keller-Segel models coupled to fluid equations,, Comm. Partial Differential Equations, 39 (2014), 1205.  doi: 10.1080/03605302.2013.852224.  Google Scholar

[3]

A. Chertock, K. Fellner, A. Kurganov, A. Lorz and P. A. Markowich, Sinking, merging and stationary plumes in a coupled chemotaxis-fluid model: a high-resolution numerical approach,, J. Fluid Mech., 694 (2012), 155.  doi: 10.1017/jfm.2011.534.  Google Scholar

[4]

R. 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]

R. Duan and Z. Xiang, A note on global existence for the chemotaxis-Stokes model with nonlinear diffusion,, Int. Math. Res. Not. IMRN, (2014), 1833.   Google Scholar

[6]

M. Di Francesco, A. Lorz and P. Markowich, Chemotaxis-fluid coupled model for swimming bacteria with nonlinear diffusion: global existence and asymptotic behavior,, Discrete Contin. Dyn. Syst., 28 (2010), 1437.  doi: 10.3934/dcds.2010.28.1437.  Google Scholar

[7]

M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 24 (1997), 633.   Google Scholar

[8]

T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis,, J. Math. Biol., 58 (2009), 183.  doi: 10.1007/s00285-008-0201-3.  Google Scholar

[9]

S. Ishida, K. Seki and T. Yokota, Boundedness in quasilinear Keller-Segel systems of parabolic-parabolic type on non-convex bounded domains,, J. Differential Equations, 256 (2014), 2993.  doi: 10.1016/j.jde.2014.01.028.  Google Scholar

[10]

S. Ishida and T. Yokota, Global existence of weak solutions to quasilinear degenerate Keller-Segel systems of parabolic-parabolic type,, J. Differential Equations, 252 (2012), 1421.  doi: 10.1016/j.jde.2011.02.012.  Google Scholar

[11]

S. Ishida and T. Yokota, Blow-up in finite or infinite time for quasilinear degenerate Keller-Segel systems of parabolic-parabolic type,, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2569.  doi: 10.3934/dcdsb.2013.18.2569.  Google Scholar

[12]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability,, J. Theor. Biol., 26 (1970), 399.  doi: 10.1016/0022-5193(70)90092-5.  Google Scholar

[13]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type,, American Mathematical Society, (1968).   Google Scholar

[14]

T. Li, A. Suen, C. Xue and M. Winkler, Global small-data solutions of a two-dimensional chemotaxis system with rotational flux term,, Math. Models Methods Appl. Sci., 25 (2015), 721.  doi: 10.1142/S0218202515500177.  Google Scholar

[15]

J.-G. Liu and A. Lorz, A coupled chemotaxis-fluid model: Global existence,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 643.  doi: 10.1016/j.anihpc.2011.04.005.  Google Scholar

[16]

J. López-Gómez, T. cc and T. Yamada, Non-trivial $\omega$-limit sets and oscillating solutions in a chemotaxis model in $\mathbbR^2$ with critical mass,, J. Funct. Anal., 266 (2014), 3455.  doi: 10.1016/j.jfa.2014.01.015.  Google Scholar

[17]

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

[18]

A. Lorz, A coupled Keller-Segel-Stokes model: Global existence for small initial data and blow-up delay,, Commun. Math. Sci., 10 (2012), 555.  doi: 10.4310/CMS.2012.v10.n2.a7.  Google Scholar

[19]

T. Nagai, T. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis,, Funkcial. Ekvac., 40 (1997), 411.   Google Scholar

[20]

K. Osaki and A. Yagi, Finite dimensional attractor for one-dimensional Keller-Segel equations,, Funkcial. Ekvac., 44 (2001), 441.   Google Scholar

[21]

Y. Seki, Y. Sugiyama and J. J. L. Velázquez, Multiple peak aggregations for the Keller-Segel system,, Nonlinearity, 26 (2013), 319.  doi: 10.1088/0951-7715/26/2/319.  Google Scholar

[22]

Y. Shibata and S. Shimizu, On the $L^p$-$L^q$ maximal regularity of the Neumann problem for the Stokes equations in a bounded domain,, J. Reine Angew. Math., 615 (2008), 157.  doi: 10.1515/CRELLE.2008.013.  Google Scholar

[23]

H. Sohr, The Navier-Stokes Equations: An Elementary Functional Analytic Approach,, Birkhäuser-Verlag, (2001).   Google Scholar

[24]

Y. Sugiyama, Global existence in sub-critical cases and finite time blow-up in super-critical cases to degenerate Keller-Segel systems,, Differential Integral Equations, 19 (2006), 841.   Google Scholar

[25]

Y. Sugiyama and Y. Yahagi, Extinction, decay and blow-up for Keller-Segel systems of fast diffusion type,, J. Differential Equations, 250 (2011), 3047.  doi: 10.1016/j.jde.2011.01.016.  Google Scholar

[26]

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

[27]

Y. Tao and M. Winkler, Global existence and boundedness in a Keller-Segel-Stokes model with arbitrary porous medium diffusion,, Discrete Contin. Dyn. Syst., 32 (2012), 1901.  doi: 10.3934/dcds.2012.32.1901.  Google Scholar

[28]

Y. Tao and M. Winkler, Locally bounded global solutions in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 157.  doi: 10.1016/j.anihpc.2012.07.002.  Google Scholar

[29]

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis,, North-Holland Publishing Co., (1977).   Google Scholar

[30]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Second edition,, Applied Mathematical Sciences, (1997).  doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[31]

I. Tuval, L. Cisneros, 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

[32]

M. Winkler, Global large-data solutions in a chemotaxis-(Navier-)Stokes system modeling cellular swimming in fluid drops,, Comm. Partial Differential Equations, 37 (2012), 319.  doi: 10.1080/03605302.2011.591865.  Google Scholar

[33]

M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system,, J. Math. Pures Appl. 100 (2013), 100 (2013), 748.  doi: 10.1016/j.matpur.2013.01.020.  Google Scholar

[34]

M. Winkler, Stabilization in a two-dimensional chemotaxis-Navier-Stokes system,, Arch. Ration. Mech. Anal. 211 (2014), 211 (2014), 455.  doi: 10.1007/s00205-013-0678-9.  Google Scholar

[35]

C. Xue and H. G. Othmer, Multiscale models of taxis-driven patterning in bacterial population,, SIAM J. Appl. Math., 70 (2009), 133.  doi: 10.1137/070711505.  Google Scholar

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