Hyperbolic conservation laws with stiff reaction terms of monostable type
Author:
Haitao Fan
Journal:
Trans. Amer. Math. Soc. 353 (2001), 41394154
MSC (2000):
Primary 35L65, 35B40, 35B25
Published electronically:
June 1, 2001
MathSciNet review:
1837224
Fulltext PDF Free Access
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Abstract: In this paper the zero reaction limit of the hyperbolic conservation law with stiff source term of monostable type
is studied. Solutions of Cauchy problems of the above equation with initial value are proved to converge, as , to piecewise constant functions. The constants are separated by either shocks determined by the RankineHugoniot jump condition, or a nonshock jump discontinuity that moves with speed . The analytic tool used is the method of generalized characteristics. Sufficient conditions for the existence and nonexistence of traveling waves of the above system with viscosity regularization are given. The reason for the failure to capture the correct shock speed by first order shock capturing schemes when underresolving is found to originate from the behavior of traveling waves of the above system with viscosity regularization.
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 W.Z. Bao and S. Jin, The random projection method for hyperbolic systems with stiff reaction terms, J. Comp. Phys. 163, 2000, 216248. CMP 2000:17
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 M. Bramson, Convergence of solutions of Kolmogorov equation to traveling waves, Memoirs AMS, vol. 44, No 285 (1983). MR 84m:60098
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 P. Colella, A. Majda and V. Roytburd, Theoretical and numerical structure for reacting shock waves, SIAM J. Sci. Stat. Comp. 7, 1986, 10591080. MR 87i:76037
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 C. M. Dafermos, Generalized characteristics and structure of solutions of hyperbolic conservation laws, Indiana Univ. Math. J. 26, 10971119, 1977. MR 56:16151
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 H. Fan, Traveling waves, Riemann problems and computations of a model of the dynamics of liquid/vapor phase transitions, J. Diff. Eqs., 150 (1998) 385437. MR 99j:35131
 [FH1]
 H. Fan and J. K. Hale, Large time behavior in inhomogeneous conservation laws, Arch. Rational Mech. Anal., 125, 201216, 1993. MR 94k:35187
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 H. Fan and J. K. Hale, Attractors in inhomogeneous conservation laws and parabolic regularizations, Trans. Ameri. Math. Soc. 347, 12391254, 1995. MR 95g:35114
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 J. Härterich, Heteroclinic orbits between rotating waves in hyperbolic balance laws, Proc. Roy. Soc. Edinburgh Sect. A, 129 (1999), 519538. MR 2000e:35136
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 S.N. Kruzhkov, First order quasilinear equations in several independent variables, Math. USSRSb. 10, 217243, 1970.
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 A. N. Lyberopoulos, A PoincarèBendixson theorem for scalar conservation laws, Proc. Roy. Soc. Edinburgh, 124A, 589607, 1994.
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 C. Sinestrari, Asymptotic profile of solutions of conservation laws with source, Differential Integral Equations 9, 499525, 1996. MR 96m:35206
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Additional Information
Haitao Fan
Affiliation:
Department of Mathematics, Georgetown University, Washington, D.C. 20057
Email:
fan@math.georgetown.edu
DOI:
http://dx.doi.org/10.1090/S0002994701027611
PII:
S 00029947(01)027611
Keywords:
Conservation law with source term,
reactionconvectiondiffusion equation,
zero reaction time limit
Received by editor(s):
October 28, 1999
Received by editor(s) in revised form:
June 19, 2000
Published electronically:
June 1, 2001
Additional Notes:
Research supported in part by NSF grant No. DMS 9705732
Article copyright:
© Copyright 2001
American Mathematical Society
