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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), 708-716. doi: 10.1126/science.153.3737.708.  Google Scholar

[2]

J. Adler, Chemoreceptors in bacteria, Science, 166 (1969), 1588-1597. 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), 767-783.  Google Scholar

[4]

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.  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), 629-641.  Google Scholar

[6]

T. Hillen and A. Potapov, The one-dimensional chemotaxis model: global existence and asymptotic profile, Math. Methods Appl. Sci., 27 (2004), 1783-1801. 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), 237-268. 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), 235-248. 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), 683-730. 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), 195-238. 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), 1967-1998. 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), 1310-1333. 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), 169-184. 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), 1044-1081. 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), 331-352. 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), 405-426. doi: 10.1002/mma.212.  Google Scholar

[17]

J. Smoller, "Shock Waves and Reaction-Diffusion Equations," 2nd edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 258, Springer-Verlag, New York, 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), 1256-1298. 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), 479-541. 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), 763-785. 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), 1017-1027. doi: 10.1090/S0002-9939-06-08773-9.  Google Scholar

show all references

References:
[1]

J. Adler, Chemotaxis in bacteria, Science, 153 (1966), 708-716. doi: 10.1126/science.153.3737.708.  Google Scholar

[2]

J. Adler, Chemoreceptors in bacteria, Science, 166 (1969), 1588-1597. 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), 767-783.  Google Scholar

[4]

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.  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), 629-641.  Google Scholar

[6]

T. Hillen and A. Potapov, The one-dimensional chemotaxis model: global existence and asymptotic profile, Math. Methods Appl. Sci., 27 (2004), 1783-1801. 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), 237-268. 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), 235-248. 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), 683-730. 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), 195-238. 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), 1967-1998. 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), 1310-1333. 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), 169-184. 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), 1044-1081. 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), 331-352. 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), 405-426. doi: 10.1002/mma.212.  Google Scholar

[17]

J. Smoller, "Shock Waves and Reaction-Diffusion Equations," 2nd edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 258, Springer-Verlag, New York, 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), 1256-1298. 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), 479-541. 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), 763-785. 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), 1017-1027. doi: 10.1090/S0002-9939-06-08773-9.  Google Scholar

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