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

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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

Abstract
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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 and 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.
[2]	J. Adler, Chemoreceptors in bacteria, Science, 166 (1969), 1588-1597. doi: 10.1126/science.166.3913.1588.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[13]	T. Nagai and T. Ikeda, Traveling waves in a chemotaxis model, J. Math. Biol., 30 (1991), 169-184. doi: 10.1007/BF00160334.
[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.
[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.
[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.
[17]	J. Smoller, "Shock Waves and Reaction-Diffusion Equations," 2^nd edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 258, Springer-Verlag, New York, 1994.
[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.
[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.
[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.
[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.

show all references

References:

[1]	J. Adler, Chemotaxis in bacteria, Science, 153 (1966), 708-716. doi: 10.1126/science.153.3737.708.
[2]	J. Adler, Chemoreceptors in bacteria, Science, 166 (1969), 1588-1597. doi: 10.1126/science.166.3913.1588.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[13]	T. Nagai and T. Ikeda, Traveling waves in a chemotaxis model, J. Math. Biol., 30 (1991), 169-184. doi: 10.1007/BF00160334.
[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.
[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.
[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.
[17]	J. Smoller, "Shock Waves and Reaction-Diffusion Equations," 2^nd edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 258, Springer-Verlag, New York, 1994.
[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.
[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.
[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.
[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.

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