Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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ASYMPTOTIC BEHAVIORS OF SOLUTIONS FOR AN AEROTAXIS MODEL COUPLED TO FLUID EQUATIONS

  • CHAE, MYEONGJU (DEPARTMENT OF APPLIED MATHEMATICS HANKYONG NATIONAL UNIVERSITY) ;
  • KANG, KYUNGKEUN (DEPARTMENT OF MATHEMATICS YONSEI UNIVERSITY) ;
  • LEE, JIHOON (DEPARTMENT OF MATHEMATICS CHUNG-ANG UNIVERSITY)
  • Received : 2014.09.11
  • Published : 2016.01.01
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Abstract

We consider a coupled system of Keller-Segel type equations and the incompressible Navier-Stokes equations in spatial dimension two. We show temporal decay estimates of solutions with small initial data and obtain their asymptotic profiles as time tends to infinity.

Keywords

References

  1. E. A. Carlen and M. Loss, Optimal smoothing and decay estimates for viscously damped conservation laws, with applications to the 22-D Navier-Stokes equation, Duke Math. J. 81 (1995), no. 1, 135-157. https://doi.org/10.1215/S0012-7094-95-08110-1
  2. A. Carpio, Asymptotic behavior for the vorticity equations in dimensions two and three, Comm. Partial Differential Equations 19 (1994), no. 5-6, 827-872. https://doi.org/10.1080/03605309408821037
  3. M. Chae, K. Kang, and J. Lee, Existence of smooth solutions to coupled chemotaxis-fluid equations, Discrete Cont. Dyn. Syst. A 33 (2013), no. 6, 2271-2297. https://doi.org/10.3934/dcds.2013.33.2271
  4. M. Chae, Global existence and temporal decay in Keller-Segel models coupled to fiuid equations, Comm. Partial Diff. Equations 39 (2014), no. 7, 1205-1235. https://doi.org/10.1080/03605302.2013.852224
  5. 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), 155-190. https://doi.org/10.1017/jfm.2011.534
  6. Y.-S. Chung, K. Kang, and J. Kim, Global existence of weak solutions for a Keller-Segel- fluid model with nonlinear diffusion, J. Korean Math. Soc. 51 (2014), no. 3, 635-654. https://doi.org/10.4134/JKMS.2014.51.3.635
  7. R. Duan, A. Lorz, and P. Markowich, Global solutions to the coupled chemotaxis-fluid equations, Comm. Partial Differential Equations 35 (2010), no. 9, 1635-1673. https://doi.org/10.1080/03605302.2010.497199
  8. M. D. Francesco, A. Lorz, and P. Markowich, Chemotaxis-fluid coupled model for swim- ming bacteria with nonlinear diffusion: global existence and asymptotic behavior, Discrete Cont. Dyn. Syst. A 28 (2010), no. 4, 1437-1453. https://doi.org/10.3934/dcds.2010.28.1437
  9. T. Gallay and C. E. Wayne, Global stability of vortex solutions of the two dimensional Navier-Stokes equation, Comm. Math. Phys. 255 (2005), no. 1, 97-129. https://doi.org/10.1007/s00220-004-1254-9
  10. M. Giga, Y. Giga, and J. Saal, Nonlinear Partial Differential Equations, Asymptotic Behavior of Solutions and Self-Similar Solutions, Birkhauser Boston, 2010.
  11. M. Giga and T. Kambe, Large time behavior of the vorticity of two-dimensional viscous flow and its application to vortex formation, Comm. Math. Phys. 117 (1988), no. 4, 549-568. https://doi.org/10.1007/BF01218384
  12. M. A. Herrero and J. L. L. Velazquez, A blow-up mechanism for a chemotaxis model, Ann. Sc. Norm. Super. Pisa Cl. Ser. 24 (1997), no. 4, 633-683.
  13. D. Horstman, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences I, Jahresber. Deutsch. Math.-Verein. 105 (2003), no. 3, 103-165.
  14. D. Horstman, From 1970 until present: The Keller-Segel model in chemotaxis and its conse- quences II, Jahresber. Deutsch. Math.-Verein. 106 (2004), no. 2, 51-69.
  15. W. Jager and S. Luckhaus, On explosions of solutions to s system of partial differential equations modeling chemotaxis, Trans. Amer. Math. Soc. 329 (1992), no. 2, 819-824. https://doi.org/10.1090/S0002-9947-1992-1046835-6
  16. E. F. Keller and L. A. Segel, Initiation of slide mold aggregation viewd as an instability, J. Theor. Biol. 26 (1970), no. 3, 399-415. https://doi.org/10.1016/0022-5193(70)90092-5
  17. E. F. Keller, Model for chemotaxis, J. Theor. Biol. 30 (1971), no. 2, 225-234. https://doi.org/10.1016/0022-5193(71)90050-6
  18. A. Lorz, Coupled chemotaxis fluid model, Math. Models Methods Appl. Sci. 20 (2010), no. 6, 987-1004. https://doi.org/10.1142/S0218202510004507
  19. T. Nagai, T. Senba, and K. Yoshida, Applications of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial Ekvac. 40 (1997), no. 3, 411-433.
  20. Y. Naito, Asymptotically self-similar solutions for the parabolic system modelling chemotaxis, Banach center publications, 74, 149-160, 2006.
  21. K. Osaki and A. Yagi, Finite dimensional attractors for one-dimensional Keller-Segel equations, Funkcial. Ekvac. 44 (2001), no. 3, 441-469.
  22. C. S. Patlak, Random walk with persistence and external bias, Bull. Math. Biophys. 15 (1953), 311-338. https://doi.org/10.1007/BF02476407
  23. R. Schweyer, Stable blow-up dynamic for the parabolic-parabolic Patlak-Keller-Segel model, arXiv:1403.4975v1.
  24. Y. Tao and M. Winkler, Global existence and boundedness in a Keller-Segel-Stokes model with arbitrary porous medium diffusion, Discrete Cont. Dyn. Syst. A 32 (2012), no. 5, 1901-1914. https://doi.org/10.3934/dcds.2012.32.1901
  25. I. Tuval, L. Cisneros, C. Dombrowski, C. W. Wolgemuth, J. O. Kessler, and R. E. Goldstein, Bacterial swimming and oxygen transport near contact lines, PNAS 102 (2005), no. 7, 2277-2282. https://doi.org/10.1073/pnas.0406724102
  26. M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations 248 (2012), no. 12, 2889-2995. https://doi.org/10.1016/j.jde.2010.02.008
  27. M. Winkler, Global large data solutions in a chemotaxis-(Navier-)Stokes system modeling cellular swimming in fluid drops, Comm. Partial Differential Equations 37 (2012), no. 2, 319-351. https://doi.org/10.1080/03605302.2011.591865
  28. M. Winkler, Stabilization in a two-dimensional chemotaxis-Navier-Stokes system, Arch. Ration. Mech. Anal. 211 (2014), no. 2, 455-487. https://doi.org/10.1007/s00205-013-0678-9

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