American Institute of Mathematical Sciences

Advanced
September  2012, 5(3): 563-581. doi: 10.3934/krm.2012.5.563

Convergence rates of zero diffusion limit on large amplitude solution to a conservation laws arising in chemotaxis

Hongyun Peng 1, , Lizhi Ruan 1, and  Changjiang Zhu 2,

1. 

The Hubei Key Laboratory of Mathematical Physics, School of Mathematics and Statistics, Central China Normal University, Wuhan, 430079, P. R., China, China
2. 

The Hubei Key Laboratory of Mathematical Physics, School of Mathematics and Statistics, Central China Normal University, Wuhan 430079

Received  January 2012 Revised  March 2012 Published  August 2012

In this paper, we investigate large amplitude solutions to a system of conservation laws which is transformed, by a change of variable, from the well-known Keller-Segel model describing cell (bacteria) movement toward the concentration gradient of the chemical that is consumed by the cells. For the Cauchy problem and initial-boundary value problem, the global unique solvability is proved based on the energy method. In particular, our main purpose is to investigate the convergence rates as the diffusion parameter $\varepsilon$ goes to zero. It is shown that the convergence rates in $L^\infty$-norm are of the order $O\left(\varepsilon\right)$ and $O(\varepsilon^{1/2})$ corresponding to the Cauchy problem and the initial-boundary value problem respectively.
Keywords: large amplitude solution, zero diffusion limit., Conservation laws, convergence rate, chemotaxis.
Mathematics Subject Classification: Primary: 92C17, 35M99; Secondary: 35Q99, 35L6.
Citation: Hongyun Peng, Lizhi Ruan, Changjiang Zhu. Convergence rates of zero diffusion limit on large amplitude solution to a conservation laws arising in chemotaxis. Kinetic & Related Models, 2012, 5 (3) : 563-581. doi: 10.3934/krm.2012.5.563
References:
[1]

J. Adler, Chemotaxis in bacteria,, Science, 153 (1966), 708. doi: 10.1126/science.153.3737.708. Google Scholar

[2]

J. Adler, Chemoreceptors in bacteria,, Science, 166 (1969), 1588. doi: 10.1126/science.166.3913.1588. Google Scholar

[3]

K. M. Chen and C. J. Zhu, The zero diffusion limit for nonlinear hyperbolic system with damping and diffusion,, J. Hyperbolic Differ. Equ., 5 (2008), 767. Google Scholar

[4]

H. Frid and V. Shelukhin, Boundary layers for the Navier-Stokes equations of compressible fluids,, Comm. Math. Phys., 208 (1999), 309. doi: 10.1007/s002200050760. Google Scholar

[5]

J. Guo, J. X. Xiao, H. J. Zhao and C. J. Zhu, Global solutions to a hyperbolic-parabolic coupled system with large initial data,, Acta Math. Sci. Ser. B Engl. Ed., 29 (2009), 629. Google Scholar

[6]

T. Hillen and A. Potapov, The one-dimensional chemotaxis model: global existence and asymptotic profile,, Math. Methods Appl. Sci., 27 (2004), 1783. doi: 10.1002/mma.569. Google Scholar

[7]

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. doi: 10.1137/07070005X. Google Scholar

[8]

E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: a theoretical analysis,, J. Theoret. Biol., 30 (1971), 235. doi: 10.1016/0022-5193(71)90051-8. Google Scholar

[9]

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. doi: 10.1137/S0036139995291106. Google Scholar

[10]

H. A. Levine, B. D. Sleeman and M. Nilsen-Hamilton, Mathematical modeling of the onset of capillary formation initating angiogenesis,, J. Math. Biol., 42 (2001), 195. doi: 10.1007/s002850000037. Google Scholar

[11]

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. doi: 10.1142/S0218202510004830. Google Scholar

[12]

T. Li and Z.-A. Wang, Asymptotic nonlinear stability of traveling waves to conservation laws arising from chemotaxis,, J. Differential Equations, 250 (2011), 1310. doi: 10.1016/j.jde.2010.09.020. Google Scholar

[13]

T. Nagai and T. Ikeda, Traveling waves in a chemotaxis model,, J. Math. Biol., 30 (1991), 169. doi: 10.1007/BF00160334. Google Scholar

[14]

H. G. Othmer and A. Stevens, Aggregation, blowup, and collapse: the ABCs of taxis in reinforced random walks,, SIAM J. Appl. Math., 57 (1997), 1044. doi: 10.1137/S0036139995288976. Google Scholar

[15]

L. Z. Ruan and C. J. Zhu, Boundary layer for nonlinear evolution equations with damping and diffusion,, Discrete Contin. Dyn. Syst., 32 (2012), 331. doi: 10.3934/dcds.2012.32.331. Google Scholar

[16]

B. D. Sleeman and H. A. Levine, Partial differential equations of chemotaxis and angiogenesis,, Math. Methods Appl. Sci., 24 (2001), 405. doi: 10.1002/mma.212. Google Scholar

[17]

J. Smoller, "Shock Waves and Reaction-Diffusion Equations,", 2nd edition, 258 (1994). Google Scholar

[18]

Y.-G. Wang and Z. P. Xin, Zero-viscosity limit of the linearized compressible Navier-Stokes equations with highly oscillatory forces in the half-plane,, SIAM J. Math. Anal., 37 (2005), 1256. doi: 10.1137/040614967. Google Scholar

[19]

Z. P. Xin and T. Yanagisawa, Zero-viscosity limit of the linearized Navier-Stokes equations for a compressible viscous fluid in the half-plane,, Comm. Pure Appl. Math., 52 (1999), 479. doi: 10.1002/(SICI)1097-0312(199904)52:4<479::AID-CPA4>3.0.CO;2-1. Google Scholar

[20]

Y. Yang, H. Chen and W. A. Liu, On existence of global solutions and blow-up to a system of reaction-diffusion equations modelling chemotaxis,, SIAM J. Math. Anal., 33 (2001), 763. doi: 10.1137/S0036141000337796. Google Scholar

[21]

M. Zhang and C. J. Zhu, Global existence of solutions to a hyperbolic-parabolic system,, Proc. Amer. Math. Soc., 135 (2007), 1017. doi: 10.1090/S0002-9939-06-08773-9. Google Scholar

show all references

References:
[1]

J. Adler, Chemotaxis in bacteria,, Science, 153 (1966), 708. doi: 10.1126/science.153.3737.708. Google Scholar

[2]

J. Adler, Chemoreceptors in bacteria,, Science, 166 (1969), 1588. doi: 10.1126/science.166.3913.1588. Google Scholar

[3]

K. M. Chen and C. J. Zhu, The zero diffusion limit for nonlinear hyperbolic system with damping and diffusion,, J. Hyperbolic Differ. Equ., 5 (2008), 767. Google Scholar

[4]

H. Frid and V. Shelukhin, Boundary layers for the Navier-Stokes equations of compressible fluids,, Comm. Math. Phys., 208 (1999), 309. doi: 10.1007/s002200050760. Google Scholar

[5]

J. Guo, J. X. Xiao, H. J. Zhao and C. J. Zhu, Global solutions to a hyperbolic-parabolic coupled system with large initial data,, Acta Math. Sci. Ser. B Engl. Ed., 29 (2009), 629. Google Scholar

[6]

T. Hillen and A. Potapov, The one-dimensional chemotaxis model: global existence and asymptotic profile,, Math. Methods Appl. Sci., 27 (2004), 1783. doi: 10.1002/mma.569. Google Scholar

[7]

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. doi: 10.1137/07070005X. Google Scholar

[8]

E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: a theoretical analysis,, J. Theoret. Biol., 30 (1971), 235. doi: 10.1016/0022-5193(71)90051-8. Google Scholar

[9]

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. doi: 10.1137/S0036139995291106. Google Scholar

[10]

H. A. Levine, B. D. Sleeman and M. Nilsen-Hamilton, Mathematical modeling of the onset of capillary formation initating angiogenesis,, J. Math. Biol., 42 (2001), 195. doi: 10.1007/s002850000037. Google Scholar

[11]

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. doi: 10.1142/S0218202510004830. Google Scholar

[12]

T. Li and Z.-A. Wang, Asymptotic nonlinear stability of traveling waves to conservation laws arising from chemotaxis,, J. Differential Equations, 250 (2011), 1310. doi: 10.1016/j.jde.2010.09.020. Google Scholar

[13]

T. Nagai and T. Ikeda, Traveling waves in a chemotaxis model,, J. Math. Biol., 30 (1991), 169. doi: 10.1007/BF00160334. Google Scholar

[14]

H. G. Othmer and A. Stevens, Aggregation, blowup, and collapse: the ABCs of taxis in reinforced random walks,, SIAM J. Appl. Math., 57 (1997), 1044. doi: 10.1137/S0036139995288976. Google Scholar

[15]

L. Z. Ruan and C. J. Zhu, Boundary layer for nonlinear evolution equations with damping and diffusion,, Discrete Contin. Dyn. Syst., 32 (2012), 331. doi: 10.3934/dcds.2012.32.331. Google Scholar

[16]

B. D. Sleeman and H. A. Levine, Partial differential equations of chemotaxis and angiogenesis,, Math. Methods Appl. Sci., 24 (2001), 405. doi: 10.1002/mma.212. Google Scholar

[17]

J. Smoller, "Shock Waves and Reaction-Diffusion Equations,", 2nd edition, 258 (1994). Google Scholar

[18]

Y.-G. Wang and Z. P. Xin, Zero-viscosity limit of the linearized compressible Navier-Stokes equations with highly oscillatory forces in the half-plane,, SIAM J. Math. Anal., 37 (2005), 1256. doi: 10.1137/040614967. Google Scholar

[19]

Z. P. Xin and T. Yanagisawa, Zero-viscosity limit of the linearized Navier-Stokes equations for a compressible viscous fluid in the half-plane,, Comm. Pure Appl. Math., 52 (1999), 479. doi: 10.1002/(SICI)1097-0312(199904)52:4<479::AID-CPA4>3.0.CO;2-1. Google Scholar

[20]

Y. Yang, H. Chen and W. A. Liu, On existence of global solutions and blow-up to a system of reaction-diffusion equations modelling chemotaxis,, SIAM J. Math. Anal., 33 (2001), 763. doi: 10.1137/S0036141000337796. Google Scholar

[21]

M. Zhang and C. J. Zhu, Global existence of solutions to a hyperbolic-parabolic system,, Proc. Amer. Math. Soc., 135 (2007), 1017. doi: 10.1090/S0002-9939-06-08773-9. Google Scholar

[1]

Freddy Dumortier. Sharp upperbounds for the number of large amplitude limit cycles in polynomial Lienard systems. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1465-1479. doi: 10.3934/dcds.2012.32.1465
[2]

Fengbai Li, Feng Rong. Decay of solutions to fractal parabolic conservation laws with large initial data. Communications on Pure & Applied Analysis, 2013, 12 (2) : 973-984. doi: 10.3934/cpaa.2013.12.973
[3]

Giuseppe Maria Coclite, Lorenzo di Ruvo, Jan Ernest, Siddhartha Mishra. Convergence of vanishing capillarity approximations for scalar conservation laws with discontinuous fluxes. Networks & Heterogeneous Media, 2013, 8 (4) : 969-984. doi: 10.3934/nhm.2013.8.969
[4]

Marcel Freitag. The fast signal diffusion limit in nonlinear chemotaxis systems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2019211
[5]

Leo G. Rebholz, Dehua Wang, Zhian Wang, Camille Zerfas, Kun Zhao. Initial boundary value problems for a system of parabolic conservation laws arising from chemotaxis in multi-dimensions. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 3789-3838. doi: 10.3934/dcds.2019154
[6]

Giuseppe Maria Coclite, Lorenzo di Ruvo. A singular limit problem for conservation laws related to the Kawahara-Korteweg-de Vries equation. Networks & Heterogeneous Media, 2016, 11 (2) : 281-300. doi: 10.3934/nhm.2016.11.281
[7]

Anupam Sen, T. Raja Sekhar. Structural stability of the Riemann solution for a strictly hyperbolic system of conservation laws with flux approximation. Communications on Pure & Applied Analysis, 2019, 18 (2) : 931-942. doi: 10.3934/cpaa.2019045
[8]

Avner Friedman. Conservation laws in mathematical biology. Discrete & Continuous Dynamical Systems - A, 2012, 32 (9) : 3081-3097. doi: 10.3934/dcds.2012.32.3081
[9]

Mauro Garavello. A review of conservation laws on networks. Networks & Heterogeneous Media, 2010, 5 (3) : 565-581. doi: 10.3934/nhm.2010.5.565
[10]

Mauro Garavello, Roberto Natalini, Benedetto Piccoli, Andrea Terracina. Conservation laws with discontinuous flux. Networks & Heterogeneous Media, 2007, 2 (1) : 159-179. doi: 10.3934/nhm.2007.2.159
[11]

Len G. Margolin, Roy S. Baty. Conservation laws in discrete geometry. Journal of Geometric Mechanics, 2019, 11 (2) : 187-203. doi: 10.3934/jgm.2019010
[12]

Marek Fila, Michael Winkler. Sharp rate of convergence to Barenblatt profiles for a critical fast diffusion equation. Communications on Pure & Applied Analysis, 2015, 14 (1) : 107-119. doi: 10.3934/cpaa.2015.14.107
[13]

Zhi-An Wang, Kun Zhao. Global dynamics and diffusion limit of a one-dimensional repulsive chemotaxis model. Communications on Pure & Applied Analysis, 2013, 12 (6) : 3027-3046. doi: 10.3934/cpaa.2013.12.3027
[14]

K. T. Joseph, Philippe G. LeFloch. Boundary layers in weak solutions of hyperbolic conservation laws II. self-similar vanishing diffusion limits. Communications on Pure & Applied Analysis, 2002, 1 (1) : 51-76. doi: 10.3934/cpaa.2002.1.51
[15]

Giuseppe Maria Coclite, Lorenzo Di Ruvo. A note on the convergence of the solution of the high order Camassa-Holm equation to the entropy ones of a scalar conservation law. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1247-1282. doi: 10.3934/dcds.2017052
[16]

Wen-Xiu Ma. Conservation laws by symmetries and adjoint symmetries. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 707-721. doi: 10.3934/dcdss.2018044
[17]

Tai-Ping Liu, Shih-Hsien Yu. Hyperbolic conservation laws and dynamic systems. Discrete & Continuous Dynamical Systems - A, 2000, 6 (1) : 143-145. doi: 10.3934/dcds.2000.6.143
[18]

Yanbo Hu, Wancheng Sheng. The Riemann problem of conservation laws in magnetogasdynamics. Communications on Pure & Applied Analysis, 2013, 12 (2) : 755-769. doi: 10.3934/cpaa.2013.12.755
[19]

Stefano Bianchini, Elio Marconi. On the concentration of entropy for scalar conservation laws. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 73-88. doi: 10.3934/dcdss.2016.9.73
[20]

Guangrong Wu, Ping Zhang. The zero diffusion limit of 2-D Navier-Stokes equations with $L^1$ initial vorticity. Discrete & Continuous Dynamical Systems - A, 1999, 5 (3) : 631-638. doi: 10.3934/dcds.1999.5.631

2018 Impact Factor: 1.38

Tools

Metrics

  • PDF downloads (4)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences