\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

Global existence for an attraction-repulsion chemotaxis fluid model with logistic source

  • * Corresponding author: Élder J. Villamizar-Roa

    * Corresponding author: Élder J. Villamizar-Roa

The second author has been partially supported by CNPq and FAPESP, Brazil. The third author has been supported by Vicerrectoría de Investigación y Extensión of Universidad Industrial de Santander, and Fondo Nacional de Financiamiento para la Ciencia, la Tecnología y la Innovación Francisco José de Caldas, contrato Colciencias FP 44842-157-2016. The authors would like to thank an anonymous referee for useful remarks and suggestions

Abstract Full Text(HTML) Related Papers Cited by
  • We consider an attraction-repulsion chemotaxis model coupled with the Navier-Stokes system. This model describes the interaction between a type of cells (e.g., bacteria), which proliferate following a logistic law, and two chemical signals produced by the cells themselves that degraded at a constant rate. Also, it is considered that the chemoattractant is consumed with a rate proportional to the amount of organisms. The cells and chemical substances are transported by a viscous incompressible fluid under the influence of a force due to the aggregation of cells. We prove the existence of global mild solutions in bounded domains of $\mathbb{R}^N,$ $N = 2, 3,$ for small initial data in $L^p$-spaces.

    Mathematics Subject Classification: Primary: 35Q92, 92C17; Secondary: 35Q35, 35K55, 76D05.

    Citation:

    \begin{equation} \\ \end{equation}
  • 加载中
  •   M. Aida , K. Osaki , T. Tsujikawa , A. Yagi  and  M. Mimura , Chemotaxis and growth system with singular sensitivity function, Nonlinear Analysis: Real World Applications, 6 (2005) , 323-336.  doi: 10.1016/j.nonrwa.2004.08.011.
      L. Angiuli , D. Pallara  and  F. Y. Paronetto , Analytic semigroups generated in L1 by second order elliptic operators via duality methods, Semigroup Forum, Springer, 80 (2010) , 255-271.  doi: 10.1007/s00233-009-9200-y.
      N. Bellomo , A. Bellouquid , Y. Tao  and  M. Winkler , Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Mathematical Models and Methods in Applied Sciences, 25 (2015) , 1663-1763.  doi: 10.1142/S021820251550044X.
      M. Braukhoff , Global (weak) solution of the chemotaxis-Navier-Stokes equations with non-homogeneous boundary conditions and logistic growth, Annales de l'Institut Henri Poincaré (C) Non Linear Analysis, 34 (2017) , 1013-1039.  doi: 10.1016/j.anihpc.2016.08.003.
      X. Cao and J. Lankeit, Global classical small-data solutions for a three-dimensional chemotaxis Navier-Stokes system involving matrix-valued sensitivities, Calc. Var. Partial Differential Equations, 55 (2016), Art. 107, 39 pp. doi: 10.1007/s00526-016-1027-2.
      T. Cazenave  and  F. B. Weissler , Asymptotically self-similar global solutions of the nonlinear Schrödinger and heat equations, Mathematische Zeitschrift, 228 (1998) , 83-120.  doi: 10.1007/PL00004606.
      S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, The International Series of Monographs on Physics Clarendon Press, Oxford, 1961.
      M.A. Chaplain  and  G. Lolas , Mathematical modelling of cancer cell invasion of tissue: The role of the urokinase plasminogen activation system, Mathematical Models and Methods in Applied Sciences, 15 (2005) , 1685-1734.  doi: 10.1142/S0218202505000947.
      M. A. Chaplain  and  A. Stuart , A model mechanism for the chemotactic response of endothelial cells to tumour angiogenesis factor, Mathematical Medicine and Biology, 10 (1993) , 149-168.  doi: 10.1093/imammb/10.3.149.
      A. Chertock , K. Fellner , A. Kurganov , A. Lorz  and  P. Markowich , Sinking, merging and stationary plumes in a coupled chemotaxis-fluid model: a high-resolution numerical approach, Journal of Fluid Mechanics, 694 (2012) , 155-190.  doi: 10.1017/jfm.2011.534.
      H. J. Choe  and  B. Lkhagvasuren , Global existence result for chemotaxis Navier-Stokes equations in the critical Besov spaces, Journal of Mathematical Analysis and Applications, 446 (2017) , 1415-1426.  doi: 10.1016/j.jmaa.2016.09.050.
      C. Dombrowski , L. Cisneros , S. Chatkaew , R. E. Goldstein  and  J. O. Kessler , Self-concentration and large-scale coherence in bacterial dynamics, Physical Review Letters, 93 (2004) , 98-103. 
      R. Duan  and  Z. Xiang , A note on global existence for the chemotaxis-Stokes model with nonlinear diffusion, International Mathematics Research Notices, 2014 (2012) , 1833-1852.  doi: 10.1093/imrn/rns270.
      E. Espejo  and  T. Suzuki , Reaction terms avoiding aggregation in slow fluids, Nonlinear Analysis: Real World Applications, 21 (2015) , 110-126.  doi: 10.1016/j.nonrwa.2014.07.001.
      L. C. F. Ferreira  and  E. J. Villamizar-Roa , Self-similar solutions, uniqueness and long-time asymptotic behavior for semilinear heat equations, Differential and Integral Equations, 19 (2006) , 1349-1370. 
      L. C. F. Ferreira  and  E. J. Villamizar-Roa , Well-posedness and asymptotic behaviour for the convection problem, Nonlinearity, 19 (2006) , 2169-2191.  doi: 10.1088/0951-7715/19/9/011.
      D. Fujiwara  and  H. Morimoto , An Lr-theorem of the Helmholtz decomposition of vector fields, IA Math, 24 (1977) , 685-700. 
      Y. Giga , Analyticity of the semigroup generated by the Stokes operator in Lr spaces, Mathematische Zeitschrift, 178 (1981) , 297-329.  doi: 10.1007/BF01214869.
      N. A. Hill  and  T. J. Pedley , Bioconvection, Fluid Dynamics Research, 37 (2005) , 1-20.  doi: 10.1016/j.fluiddyn.2005.03.002.
      T. Hillen  and  K. J. Painter , A user's guide to PDE models for chemotaxis, Journal of Mathematical Biology, 58 (2009) , 183-217.  doi: 10.1007/s00285-008-0201-3.
      T. Hillen , K. J. Painter  and  M. Winkler , Convergence of a cancer invasion model to a logistic chemotaxis model, Mathematical Models and Methods in Applied Sciences, 23 (2013) , 165-198.  doi: 10.1142/S0218202512500480.
      D. Horstmann , Generalizing the Keller-Segel model: Lyapunov functionals, steady state analysis, and blow-up results for multi-species chemotaxis models in the presence of attraction and repulsion between competitive interacting species, Journal of Nonlinear Science, 21 (2011) , 231-270.  doi: 10.1007/s00332-010-9082-x.
      S. Ishida , Global existence and boundedness for chemotaxis-Navier-Stokes systems with position-dependent sensitivity in 2D bounded domains, Discrete & Continuous Dynamical Systems-A, 35 (2015) , 3463-3482.  doi: 10.3934/dcds.2015.35.3463.
      J. Jiang , H. Wu  and  S. Zheng , Global existence and asymptotic behavior of solutions to a chemotaxis-fluid system on general bounded domains, Asymptotic Analysis, 92 (2015) , 249-258. 
      T. Kato , Strong Lp-solutions of the Navier-Stokes equation in Rm with applications to weak solutions, Math. Z., 187 (1984) , 471-480.  doi: 10.1007/BF01174182.
      T. Kato , Strong solutions of the Navier-Stokes equation in Morrey spaces, Bol. Soc. Brasil. Mat., 22 (1992) , 127-155.  doi: 10.1007/BF01232939.
      A. Kiselev  and  L. Ryzhik , Biomixing by chemotaxis and enhancement of biological reactions, Communications in Partial Differential Equations, 37 (2012) , 298-318.  doi: 10.1080/03605302.2011.589879.
      H. Kozono , M. Miura  and  Y. Sugiyama , Existence and uniqueness theorem on mild solutions to the Keller-Segel system coupled with the Navier-Stokes fluid, Journal of Functional Analysis, 270 (2016) , 1663-1683.  doi: 10.1016/j.jfa.2015.10.016.
      J. Lankeit , Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source, Journal of Differential Equations, 258 (2015) , 1158-1191.  doi: 10.1016/j.jde.2014.10.016.
      J. Lankeit , Long-term behaviour in a chemotaxis-fluid system with logistic source, Math. Models Methods Appl. Sci., 26 (2016) , 2071-2109.  doi: 10.1142/S021820251640008X.
      D. Li , C. Mu , K. Lin  and  L. Wang , Large time behavior of solution to an attraction-repulsion chemotaxis system with logistic source in three dimensions, Journal of Mathematical Analysis and Applications, 448 (2016) , 914-936.  doi: 10.1016/j.jmaa.2016.11.036.
      X. Li , Boundedness in a two-dimensional attraction-repulsion system with nonlinear diffusion, Mathematical Methods in the Applied Sciences, 39 (2016) , 289-301.  doi: 10.1002/mma.3477.
      X. Li  and  Z. Xiang , On an attraction-repulsion chemotaxis system with a logistic source, IMA Journal of Applied Mathematics, 81 (2016) , 165-198.  doi: 10.1093/imamat/hxv033.
      J. Liu  and  Y. Wang , Global existence and boundedness in a Keller-Segel-(Navier-) Stokes system with signal-dependent sensitivity, Journal of Mathematical Analysis and Applications, 447 (2017) , 499-528.  doi: 10.1016/j.jmaa.2016.10.028.
      J. Liu  and  Y. Wang , Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system involving a tensor-valued sensitivity with saturation, Journal of Differential Equations, 262 (2017) , 5271-5305.  doi: 10.1016/j.jde.2017.01.024.
      P. Liu , J. Shi  and  Z.-A. Wang , Pattern formation of the attraction-repulsion Keller-Segel system, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013) , 2597-2625.  doi: 10.3934/dcdsb.2013.18.2597.
      M. Luca , A. Chavez-Ross , L. Edelstein-Keshet  and  A. Mogilner , Chemotactic signaling, microglia, and Alzheimer's disease senile plaques: Is there a connection?, Bulletin of Mathematical Biology, 65 (2003) , 693-730.  doi: 10.1016/S0092-8240(03)00030-2.
      A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser/Springer Basel AG, Basel, 1995.
      N. V. Mantzaris , S. Webb  and  H. G. Othmer , Mathematical modeling of tumor-induced angiogenesis, Journal of Mathematical Biology, 49 (2004) , 111-187.  doi: 10.1007/s00285-003-0262-2.
      X. Mora , Semilinear parabolic problems define semiflows on Ck spaces, Transactions of the American Mathematical Society, 278 (1983) , 21-55.  doi: 10.2307/1999300.
      A. Quinlan  and  B. Straughan , Decay bounds in a model for aggregation of microglia: Application to Alzheimer's disease senile plaques, Proceedings of the Royal Society A, 461 (2005) , 2887-2897.  doi: 10.1098/rspa.2005.1483.
      Y. Tao  and  M. Winkler , Locally bounded global solutions in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion, Annales de l'Institut Henri Poincaré (C) Non Linear Analysis, 30 (2013) , 157-178.  doi: 10.1016/j.anihpc.2012.07.002.
      Y. Tao  and  M. Winkler , Boundedness and decay enforced by quadratic degradation in a three-dimensional chemotaxis-fluid system, Zeitschrift für angewandte Mathematik und Physik, 66 (2015) , 2555-2573.  doi: 10.1007/s00033-015-0541-y.
      Y. Tao and M. Winkler, Blow-up prevention by quadratic degradation in a two-dimensional Keller-Segel-Navier-Stokes system Z. Angew. Math. Phys., 67 (2016), Art. 138, 23 pp. doi: 10.1007/s00033-016-0732-1.
      J. I. Tello  and  M. Winkler , A chemotaxis system with logistic source, Communications in Partial Differential Equations, 32 (2007) , 849-877.  doi: 10.1080/03605300701319003.
      I. Tuval , L. Cisneros , C. Dombrowski , C. W. Wolgemuth , J. O. Kessler  and  R. E. Goldstein , Bacterial swimming and oxygen transport near contact lines, Proceedings of the National Academy of Sciences of the United States of America, 102 (2005) , 2277-2282.  doi: 10.1073/pnas.0406724102.
      R. Tyson , S. R. Lubkin  and  J. D. Murray , Model and analysis of chemotactic bacterial patterns in a liquid medium, Journal of Mathematical Biology, 38 (1999) , 359-375.  doi: 10.1007/s002850050153.
      Y. Wang , Boundedness in a three-dimensional attraction-repulsion chemotaxis system with nonlinear diffusion and logistic source, Electronic Journal of Differential Equations, 176 (2016) , 1-21. 
      Y. Wang  and  Z. Xiang , Global existence and boundedness in a Keller-Segel-Stokes system involving a tensor-valued sensitivity with saturation: The 3D case, Journal of Differential Equations, 261 (2016) , 4944-4973.  doi: 10.1016/j.jde.2016.07.010.
      M. Winkler , Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, Journal of Differential Equations, 248 (2010) , 2889-2905.  doi: 10.1016/j.jde.2010.02.008.
      M. Winkler , Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Communications in Partial Differential Equations, 35 (2010) , 1516-1537.  doi: 10.1080/03605300903473426.
      M. Winkler , Global large-data solutions in a chemotaxis-(Navier-) Stokes system modeling cellular swimming in fluid drops, Communications in Partial Differential Equations, 37 (2012) , 319-351.  doi: 10.1080/03605302.2011.591865.
      M. Winkler , Global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with strong logistic dampening, Journal of Differential Equations, 257 (2014) , 1056-1077.  doi: 10.1016/j.jde.2014.04.023.
      M. Winkler , Boundedness and large time behavior in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion and general sensitivity, Calculus of Variations and Partial Differential Equations, 54 (2015) , 3789-3828.  doi: 10.1007/s00526-015-0922-2.
      M. Winkler , Global weak solutions in a three-dimensional chemotaxis-Navier-Stokes system, Annales de l'Institut Henri Poincaré (C) Non Linear Analysis, 33 (2016) , 1329-1352.  doi: 10.1016/j.anihpc.2015.05.002.
      D. Woodward, R. Tyson, M. Myerscough, J. D. Murray, E. Budrene, and H. Berg, Spatio-temporal patterns generated by Salmonella typhimurium Biophysical Journal, 68 (1995), no. 5, 2181. doi: 10.1016/S0006-3495(95)80400-5.
      Q. Zhang  and  Y. Li , An attraction-repulsion chemotaxis system with logistic source, Biophysical Journal, 96 (2015) , 570-584.  doi: 10.1002/zamm.201400311.
      Q. Zhang  and  Y. Li , Global weak solutions for the three-dimensional chemotaxis-Navier-Stokes system with nonlinear diffusion, Journal of Differential Equations, 259 (2015) , 3730-3754.  doi: 10.1016/j.jde.2015.05.012.
      P. Zheng , C. Mu  and  X. Hu , Boundedness in the higher dimensional attraction-repulsion chemotaxis-growth system, Computers & Mathematics with Applications, 72 (2016) , 2194-2202.  doi: 10.1016/j.camwa.2016.08.028.
  • 加载中
SHARE

Article Metrics

HTML views(1980) PDF downloads(338) Cited by(0)

Access History

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return