Acta mathematica scientia, Series B >
ON THE CONVERGENCE RATE OF A CLASS OF REACTION HYPERBOLIC SYSTEMS FOR AXONAL TRANSPORT
Received date: 2015-03-23
Online published: 2015-07-01
Supported by
The work of F. Huang is partially supported by the NSFC (11371349), National Basic Research Program of China (973 Program) (2011CB808002).
In this paper, we consider a class of reaction hyperbolic systems for axonal transport arising in neuroscience which can be regarded as hyperbolic systems with relaxation. We prove the BV entropy solutions of the hyperbolic systems converge toward to the unique entropy solution of the equilibrium equation at the optimal rate O(√δ)in L1 norm as the relaxation time δ tends to zero.
Key words： axonal transport; relaxation; equilibrium; convergence rate
Wentao CAO , Feimin HUANG . ON THE CONVERGENCE RATE OF A CLASS OF REACTION HYPERBOLIC SYSTEMS FOR AXONAL TRANSPORT[J]. Acta mathematica scientia, Series B, 2015 , 35(4) : 945 -954 . DOI: 10.1016/S0252-9602(15)30029-1
[1] Bear M, Connors B, Paradiso M. Neuroscience: Exploring the Brain. 2nd ed. Baltimore: Lippincott Williams Wilkins, 2001
[2] Blum J, Reed M. A model for fast axonal transport. Cell Motility Cytoskeleton, 1985, 5(6): 507-527
[3] Blum J, Reed M. The transport of organelles in axons. Math Biosci, 1988, 90: 233-245
[4] Carr D D. Global exsistence of solutions to reaxtion-hyperbolic systems in one space dimension. SIAM J Math Anal, 1995, 26(2): 399-414
[5] Friedman A, Craciun G. Approximate traveling waves in linear reaction-hyperbolic equations. SIAM J Math Anal, 2006, 38(3): 741-758
[6] Friedman A, Hu B. Uniform convergence for approximate traveling waves in linear reaction-hyperbolic systems. Indiana Univ Math J, 2007, 56(5): 2133-2158
[7] Hanouzet B, Natalini R. Weakly coupled systems of quasilinear hyperbolic equations. Differential Integral Equations, 1996, 9(6): 1279-1292
[8] Kandel E, et al. Principles of Neural Sciences. 4th ed. McGraw Hill, 2000
[9] Kruzkov S N. First order quasilinear euations with several space variables. Math USSRSb, 1970, 10(2): 217-243
[10] Kurganov A, Tadmor E. Stiff systems of hyperbolic conservation laws: convergence and error estimate. SIAM J Math Anal, 1997, 28(6): 1446-1456
[11] Kuznetsov N N. The accuracy of certain approximate methods for computation of weak solutions of a first order quasilinear equation. USSR Comput Math Phys, 1976, 16: 105-119
[12] Liu T -P. Hyperbolic conservation laws with relaxation. Comm Math Phys, 1987, 108(1): 153-175
[13] Reed M, Blum J. Theoretical Analysis of radioactivity profiles during fast axonal transport: Effects of deposition and turnover. Cell Motility Cytoskeleton, 1986, 6(6): 620-627
[14] Reed M, Blum J. A model for slow axonal transport and its application to neurofilamentous neuropathies. Cell Motility Cytoskeleton, 1989, 12(1): 53-65
[15] Reed M, Blum J. Mathematical questions in axonal transport//Lectures on Mathematics in the Life Sci- ences, 24. Amer Math Soc, 1994
[16] Reed M, Venakides S, Blum J. Approximate traveling waves in linear reaction-hyperbolic equations. SIAM J Appl Math, 1990, 50(1): 167-180
[17] Schroll H J, Tveito A, Winther R. An L1 error bound for a semi-implicit difference scheme apllied to a stiff system of conservation laws. SIAM J Math Anal, 1997, 34(3): 1152-1166
[18] Smoller J. Shock Waves and Rarefaction-Diffusion Equations. New York: Springer-Verlag, 1983
[19] Tang T, Teng Z H. The sharpness of Kuznetsov's O(1)(√Δx) L1-error estimate for monotone diffference schemes. Math Comp, 1995, 64: 581-589
[20] A.Tveito and R.Winther, On the rate of convergence to equilibrium for a stiff system of conservation laws with s relaxtion term. SIAM J Math Anal, 1997, 28(1): 136-161
[21] Whitham G. Linear and Nonlinear Waves. New York: Wiley-Interscience, 1974
[22] Yan H, Yong W -A. Weak entropy solutions of nonlinear reaction-hyperbolic systems for axonal transport. Math Mod Meth Appl Sci, 2011, 28(10): 2135-2154"
/
| 〈 |
|
〉 |