Advanced Search
Article Contents
Article Contents

Boundary layers and stabilization of the singular Keller-Segel system

  • * Corresponding author

    * Corresponding author 
Abstract Full Text(HTML) Figure(3) Related Papers Cited by
  • The original Keller-Segel system proposed in [23] remains poorly understood in many aspects due to the logarithmic singularity. As the chemical consumption rate is linear, the singular Keller-Segel model can be converted, via the Cole-Hopf transformation, into a system of viscous conservation laws without singularity. However the chemical diffusion rate parameter ε now plays a dual role in the transformed system by acting as the coefficients of both diffusion and nonlinear convection. In this paper, we first consider the dynamics of the transformed Keller-Segel system in a bounded interval with time-dependent Dirichlet boundary conditions. By imposing appropriate conditions on the boundary data, we show that boundary layer profiles are present as ε→0 and large-time profiles of solutions are determined by the boundary data. We employ weighted energy estimates with the "effective viscous flux" technique to establish the uniform-in-ε estimates to show the emergence of boundary layer profiles. For asymptotic dynamics of solutions, we develop a new idea by exploring the convexity of an entropy expansion to get the basic L1-estimate. We the obtain the corresponding results for the original Keller-Segel system by reversing the Cole-Hopf transformation. Numerical simulations are performed to interpret our analytical results and their implications.

    Mathematics Subject Classification: Primary: 92C17, 35Q92, 35K57; Secondary: 35A01, 35B40, 35B44.


    \begin{equation} \\ \end{equation}
  • 加载中
  • Figure 1.  Numerical simulation of the evolution of solution profiles of the system (4) as $\varepsilon\to 0$ in the interval $[0, 1]$, where $u|_{x=0, 1}=1+0.1\sin(t), v|_{x=0, 1}=1+0.1\sin(t), u_0(x)=1-\sin(\pi x), v_0(x)=1+x(1-x)$. The solution $(u(x,t), v(x,t)$ is plotted at time $t=0.2$

    Figure 2.  Numerical simulation of the time evolution of boundary layer solutions of (4) with $\varepsilon=0.0001$ in the interval $[0, 1]$, where the initial and boundary date are same as those chosen in Fig. 1

    Figure 3.  Numerical simulation of the time evolution of solutions to (4) in the interval $[0, 1]$ with decay boundary data, where $u|_{x=0,1}=1+\exp(-t), v|_{x=0, 1}=1/(1+t), u_0(x)=2+x(1-x),v_0(x)=1+x(1-x)$, and $\chi=D=1, \varepsilon=0.0001$

  • [1] J. Adler, Chemotaxis in bacteria, Science, 153 (1966), 708-716. 
    [2] W. Alt and D. A. Lauffenburger, Transient behavior of a chemotaxis system modeling certain types of tissue inflammation, J. Math. Biol., 24 (1987), 691-722.  doi: 10.1007/BF00275511.
    [3] D. Balding and D. L. S. McElwain, A mathematical model of tumour-induced capillary growth, J. Theor. Biol., 114 (1985), 53-73.  doi: 10.1016/S0022-5193(85)80255-1.
    [4] A. Bressan, Hyperbolic Systems of Conservation Laws: The One-dimensional Cauchy Problem, Oxford University Press, 2000.
    [5] M. Chae, K. Choi, K. Kang and J. Lee, Stability of planar traveling waves in a Keller-Segel equation on an infinite strip domain, Journal of Differential Equations, 265 (2018), 237-279, arXiv: 1609.00821v1. doi: 10.1016/j.jde.2018.02.034.
    [6] M. A. J. Chaplain and A. M. Stuart, A model mechanism for the chemotactic response of endothelial cells to tumor angiogenesis factor, IMA J. Math. Appl. Med., 10 (1993), 149-168. 
    [7] L. CorriasB. Perthame and H. Zaag, A chemotaxis model motivated by angiogenesis, C. R. Math. Acad. Sci. Paris, 2 (2003), 141-146.  doi: 10.1016/S1631-073X(02)00008-0.
    [8] L. CorriasB. Perthame and H. Zaag, Global solutions of some chemotaxis and angiogenesis systems in high space dimensions, Milan J. Math., 72 (2004), 1-28.  doi: 10.1007/s00032-003-0026-x.
    [9] C. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, 4th Edition, Spring-Verlag, 2016. doi: 10.1007/978-3-662-49451-6.
    [10] F. W. DahlquistP. Lovely and D. E. Jr Koshland, Qualitative analysis of bacterial migration in chemotaxis, Nature, New Biol., 236 (1972), 120-123. 
    [11] P. N. Davis, P. van Heijster and R. Marangell, Absolute instabilities of traveling wave solutions in a Keller-Segel model, arXiv: 1608.05480v2.
    [12] C. Deng and T. Li, Well-posedness of a 3D parabolic-hyperbolic Keller-Segel system in the Sobolev space framework, J. Differential Equations, 257 (2014), 1311-1332.  doi: 10.1016/j.jde.2014.05.014.
    [13] H. Frid and V. Shelukhin, Boundary layers for the Navier-Stokes equations of compressible fluids, Comm. Math. Phys., 208 (1999), 309-330.  doi: 10.1007/s002200050760.
    [14] A. Gamba, D. Ambrosi, A. Coniglio, A. de Candia, S. Di Talia, E. Giraudo, G. Serini, L. Preziosi and F. Bussolino, Percolation, morphogenesis, and Burgers dynamics in blood vessels formation, Phys. Rev. Lett., 90 (2003), 118101. doi: 10.1103/PhysRevLett.90.118101.
    [15] C. Hao, Global well-posedness for a multidimensional chemotaxis model in critical Besov spaces, Z. Angew Math. Phys., 63 (2012), 825-834.  doi: 10.1007/s00033-012-0193-0.
    [16] H. HöferJ. A. Sherratt and P. K. Maini, Cellular pattern formation during Dictyostelium aggregation, Physica D., 85 (1995), 425-444. 
    [17] D. Hoff, Global solutions of the Navier-Stokes equations for multidimensional compressible flow with discontinuous initial data, J. Differential Equations, 120 (1995), 215-254.  doi: 10.1006/jdeq.1995.1111.
    [18] Q. Q. HouZ. A. Wang and K. Zhao, Boundary layer problem on a hyperbolic system arising from chemotaxis, J. Differential Equations, 261 (2016), 5035-5070.  doi: 10.1016/j.jde.2016.07.018.
    [19] S. Jiang and J. W. Zhang, On the non-resistive limit and the magnetic boundary-layer for one-dimensional compressible magnetohydrodynamics, Nonlinearity, 30 (2017), 3587-3612. 
    [20] S. Jiang and J. W. Zhang, Boundary layers for the Navier-Stokes equations of compressible heat-conducting flows with cylindrical symmetry, SIAM J. Math. Anal., 41 (2009), 237-268.  doi: 10.1137/07070005X.
    [21] H. Y. JinJ. Y. Li and Z. A. Wang, Asymptotic stability of traveling waves of a chemotaxis model with singular sensitivity, J. Differential Equations, 255 (2013), 193-219.  doi: 10.1016/j.jde.2013.04.002.
    [22] Y. V. KalininL. JiangY. Tu and M. Wu, Logarithmic sensing in Escherichia coli bacterial chemotaxis, Biophysical J., 96 (2009), 2439-2448.  doi: 10.1016/j.bpj.2008.10.027.
    [23] E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: A theoretical analysis, J. Theor. Biol., 30 (1971), 235-248.  doi: 10.1016/0022-5193(71)90051-8.
    [24] E. F. Keller and G. M. Odell, Necessary and sufficient conditions for chemotactic bands, Math. Biosci., 27 (1976), 309-317.  doi: 10.1016/0025-5564(75)90109-1.
    [25] E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theor. Biol., 30 (1971), 225-234.  doi: 10.1016/0022-5193(71)90050-6.
    [26] H. A. Levine and B. D. Sleeman, A system of reaction diffusion equations arising in the theory of reinforced random walks, SIAM J. Appl. Math., 57 (1997), 683-730.  doi: 10.1137/S0036139995291106.
    [27] H. A. LevineB. D. Sleeman and M. Nilsen-Hamilton, A mathematical model for the roles of pericytes and macrophages in the initiation of angiogenesis. i. the role of protease inhibitors in preventing angiogenesis, Math. Biosci., 168 (2000), 77-115.  doi: 10.1016/S0025-5564(00)00034-1.
    [28] D. LiR. Pan and K. Zhao, Quantitative decay of a one-dimensional hybrid chemotaxis model with large data, Nonlinearity, 7 (2015), 2181-2210.  doi: 10.1088/0951-7715/28/7/2181.
    [29] H. Li and K. Zhao, Initial-boundary value problems for a system of hyperbolic balance laws arising from chemotaxis, J. Differential Equations, 258 (2015), 302-338.  doi: 10.1016/j.jde.2014.09.014.
    [30] J. LiT. Li and Z. A. Wang, Stability of traveling waves of the keller-segel system with logarithmic sensitivity, Math. Models Methods Appl. Sci., 24 (2014), 2819-2849.  doi: 10.1142/S0218202514500389.
    [31] T. LiR. Pan and K. Zhao, Global dynamics of a hyperbolic-parabolic model arising from chemotaxis, SIAM J. Appl. Math., 72 (2012), 417-443.  doi: 10.1137/110829453.
    [32] T. Li and Z. A. Wang, Nonlinear stability of traveling waves to a hyperbolic-parabolic system modeling chemotaxis, SIAM J. Appl. Math., 70 (2009), 1522-1541.  doi: 10.1137/09075161X.
    [33] T. Li and Z. A. Wang, Nonlinear stability of large amplitude viscous shock waves of a generalized hyperbolic-parabolic system arising in chemotaxis, Math. Models Methods Appl. Sci., 20 (2010), 1967-1998.  doi: 10.1142/S0218202510004830.
    [34] T. Li and Z. A. Wang, Asymptotic nonlinear stability of traveling waves to conservation laws arising from chemotaxis, J. Differential Equations, 250 (2011), 1310-1333.  doi: 10.1016/j.jde.2010.09.020.
    [35] T. Li and Z. A. Wang, Steadily propagating waves of a chemotaxis model, Math. Biosci., 240 (2012), 161-168.  doi: 10.1016/j.mbs.2012.07.003.
    [36] P. L. Lions, Mathematical Topics in Fluid Mechanics. Vol. Ⅱ, Compressible Models, Clarendon Press, 1998.
    [37] V. Martinez, Z. A. Wang and K. Zhao, Asymptotic and viscous stability of large-amplitude solutions of a hyperbolic system arising from biology, Indiana Univ. Math. J., to appear.
    [38] J. D. Murray, Mathematical Biology Ⅰ: An Introduction, 3rd edition, Springer, Berlin, 2002.
    [39] T. Nagai and T. Ikeda, Traveling waves in a chemotactic model, J. Math. Biol., 30 (1991), 169-184.  doi: 10.1007/BF00160334.
    [40] R. Nossal, Boundary movement of chemotactic bacterial populations, Math. Biosci., 13 (1972), 397-406.  doi: 10.1016/0025-5564(72)90058-2.
    [41] K. J. PainterP. K. Maini and H. G. Othmer, Stripe formation in juvenile pomacanthus explained by a generalized Turing mechanism with chemotaxis, Proc. Natl. Acad. Sci., 96 (1999), 5549-5554.  doi: 10.1073/pnas.96.10.5549.
    [42] K. J. PainterP. K. Maini and H. G. Othmer, A chemotactic model for the advance and retreat of the primitive streak in avian development, Bull. Math. Biol., 62 (2000), 501-525. 
    [43] H. Y. PengH. Y. Wen and C. J. Zhu, Global well-posedness and zero diffusion limit of classical solutions to 3D conservation laws arising in chemotaxis, Z. Angew Math. Phys, 65 (2014), 1167-1188.  doi: 10.1007/s00033-013-0378-1.
    [44] G. J. PetterH. M. ByrneD. L. S. McElwain and J. Norbury, A model of wound healing and angiogenesis in soft tissue, Math. Biosci., 136 (2003), 35-63. 
    [45] X. L. QinT. YangZ. A. Yao and W. S. Zhou, Vanishing shear viscosity and boundary layer for the Navier-Stokes equations with cylindrical symmetry, Arch. Ration. Mech. Anal, 216 (2015), 1049-1086.  doi: 10.1007/s00205-014-0826-x.
    [46] H. Schwetlick, Traveling waves for chemotaxis systems, Prof. Appl. Math. Mech., 3 (2003), 476-478.  doi: 10.1002/pamm.200310508.
    [47] J. Smoller, Shock Waves and Reaction-Diffusion Equations, 2nd Edition, Spring-Verlag, Berlin, 1994. doi: 10.1007/978-1-4612-0873-0.
    [48] R. TysonS. R. Lubkin and J. Murray, Model and analysis of chemotactic bacterial patterns in a liquid medium, J. Math. Biol., 38 (1999), 359-375.  doi: 10.1007/s002850050153.
    [49] Z. A. Wang, Mathematics of traveling waves in chemotaxis, Discrete Contin. Dyn. Syst - B, 18 (2013), 601-641.  doi: 10.3934/dcdsb.2013.18.601.
    [50] Z. A. WangY. S. Tao and L. H. Wang, Large-time behavior of a parabolic-parabolic chemotaxis model with logarithmic sensitivity in one dimension, Discrete Contin. Dyn. System - B, 18 (2013), 821-845. 
    [51] Z. A. WangZ. Xiang and P. Yu, Asymptotic dynamics on a singular chemotaxis system modeling onset of tumor angiogenesis, J. Differential Equations, 260 (2016), 2225-2258.  doi: 10.1016/j.jde.2015.09.063.
    [52] Z. A. Wang and K. Zhao, Global dynamics and diffusion limit of a one-dimensional repulsive chemotaxis model, Comm. Pure Appl. Anal., 12 (2013), 3027-3046.  doi: 10.3934/cpaa.2013.12.3027.
    [53] M. Winkler, The two-dimensional Keller-Segel system with singular sensitivity and signal absorption: Global large-data solutions and their relaxation properties, Math. Models Methods Appl. Sci., 26 (2016), 987-1024.  doi: 10.1142/S0218202516500238.
    [54] L. YaoT. Zhang and C. J. Zhu, Boundary layers for compressible Navier-Stokes equations with density-dependent viscosity and cylindrical symmetry, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 677-709.  doi: 10.1016/j.anihpc.2011.04.006.
  • 加载中



Article Metrics

HTML views(1529) PDF downloads(364) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint