# American Institute of Mathematical Sciences

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

 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.
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," 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.
 [1] Neng Zhu, Zhengrong Liu, Fang Wang, Kun Zhao. Asymptotic dynamics of a system of conservation laws from chemotaxis. Discrete and Continuous Dynamical Systems, 2021, 41 (2) : 813-847. doi: 10.3934/dcds.2020301 [2] Freddy Dumortier. Sharp upperbounds for the number of large amplitude limit cycles in polynomial Lienard systems. Discrete and Continuous Dynamical Systems, 2012, 32 (5) : 1465-1479. doi: 10.3934/dcds.2012.32.1465 [3] Mingying Zhong. Diffusion limit and the optimal convergence rate of the Vlasov-Poisson-Fokker-Planck system. Kinetic and Related Models, 2022, 15 (1) : 1-26. doi: 10.3934/krm.2021041 [4] Fengbai Li, Feng Rong. Decay of solutions to fractal parabolic conservation laws with large initial data. Communications on Pure and Applied Analysis, 2013, 12 (2) : 973-984. doi: 10.3934/cpaa.2013.12.973 [5] Giuseppe Maria Coclite, Lorenzo di Ruvo, Jan Ernest, Siddhartha Mishra. Convergence of vanishing capillarity approximations for scalar conservation laws with discontinuous fluxes. Networks and Heterogeneous Media, 2013, 8 (4) : 969-984. doi: 10.3934/nhm.2013.8.969 [6] Stefano Galatolo, Hugo Marsan. Quadratic response and speed of convergence of invariant measures in the zero-noise limit. Discrete and Continuous Dynamical Systems, 2021, 41 (11) : 5303-5327. doi: 10.3934/dcds.2021078 [7] Marcel Freitag. The fast signal diffusion limit in nonlinear chemotaxis systems. Discrete and Continuous Dynamical Systems - B, 2020, 25 (3) : 1109-1128. doi: 10.3934/dcdsb.2019211 [8] 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 and Continuous Dynamical Systems, 2019, 39 (7) : 3789-3838. doi: 10.3934/dcds.2019154 [9] Mingshang Hu, Xiaojuan Li, Xinpeng Li. Convergence rate of Peng’s law of large numbers under sublinear expectations. Probability, Uncertainty and Quantitative Risk, 2021, 6 (3) : 261-266. doi: 10.3934/puqr.2021013 [10] Marco Di Francesco, Graziano Stivaletta. Convergence of the follow-the-leader scheme for scalar conservation laws with space dependent flux. Discrete and Continuous Dynamical Systems, 2020, 40 (1) : 233-266. doi: 10.3934/dcds.2020010 [11] Jie Zhao. Large time behavior of solution to quasilinear chemotaxis system with logistic source. Discrete and Continuous Dynamical Systems, 2020, 40 (3) : 1737-1755. doi: 10.3934/dcds.2020091 [12] Avner Friedman. Conservation laws in mathematical biology. Discrete and Continuous Dynamical Systems, 2012, 32 (9) : 3081-3097. doi: 10.3934/dcds.2012.32.3081 [13] Mauro Garavello. A review of conservation laws on networks. Networks and Heterogeneous Media, 2010, 5 (3) : 565-581. doi: 10.3934/nhm.2010.5.565 [14] 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 [15] Mauro Garavello, Roberto Natalini, Benedetto Piccoli, Andrea Terracina. Conservation laws with discontinuous flux. Networks and Heterogeneous Media, 2007, 2 (1) : 159-179. doi: 10.3934/nhm.2007.2.159 [16] Giuseppe Maria Coclite, Lorenzo di Ruvo. A singular limit problem for conservation laws related to the Kawahara-Korteweg-de Vries equation. Networks and Heterogeneous Media, 2016, 11 (2) : 281-300. doi: 10.3934/nhm.2016.11.281 [17] 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 and Applied Analysis, 2019, 18 (2) : 931-942. doi: 10.3934/cpaa.2019045 [18] K. T. Joseph, Philippe G. LeFloch. Boundary layers in weak solutions of hyperbolic conservation laws II. self-similar vanishing diffusion limits. Communications on Pure and Applied Analysis, 2002, 1 (1) : 51-76. doi: 10.3934/cpaa.2002.1.51 [19] Zhijie Cao, Lijun Zhang. Symmetries and conservation laws of a time dependent nonlinear reaction-convection-diffusion equation. Discrete and Continuous Dynamical Systems - S, 2020, 13 (10) : 2703-2717. doi: 10.3934/dcdss.2020218 [20] Marek Fila, Michael Winkler. Sharp rate of convergence to Barenblatt profiles for a critical fast diffusion equation. Communications on Pure and Applied Analysis, 2015, 14 (1) : 107-119. doi: 10.3934/cpaa.2015.14.107

2020 Impact Factor: 1.432