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Convergence rates of zero diffusion limit on large amplitude solution to a conservation laws arising in chemotaxis
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 |
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," 2nd 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," 2nd 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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