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BV solutions and relaxation limit for a model in viscoelasticity

Published online by Cambridge University Press:  14 November 2011

Tao Luo  and
Roberto Natalini
Tao Luo
Affiliation:
Istituto per le Applicazioni del Calcolo ‘M. Picone’, Viale del Policlinico 137,1-00161 Roma, Italy
Roberto Natalini
Affiliation:
Istituto per le Applicazioni del Calcolo ‘M. Picone’, Viale del Policlinico 137,1-00161 Roma, Italy
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Abstract

Global existence of uniformly bounded BV entropy solutions to a 2 × 2 quasilinear system with relaxation, arising in viscoelasticity, is established by using special fractional step versions of Godunov and Glimm schemes for arbitrarily large initial data. Thanks to the uniform estimates obtained, we prove the convergence of solutions to the corresponding equilibrium limit as the relaxation parameter tends to zero.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 128 , Issue 4 , 1998 , pp. 775 - 795
Copyright
Copyright © Royal Society of Edinburgh 1998

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References

1Aregba-Driollet, D. and Natalini, R.. Convergence of relaxation schemes for conservation laws. Appl. Anal. 61 (1996), 163–93.CrossRefGoogle Scholar
2Chang, T. and Hsiao, L.. The Riemann Problem and Interaction of Waves in Gas Dynamics, Pitman Monographs and Surveys in Pure and Applied Mathematics 41 (Harlow: Longman Scientific and Technical, 1989).Google Scholar
3Chen, G.-Q. and Liu, T. P.. Zero relaxation and dissipation limits for hyperbolic conservation laws. Comm. Pure Appl. Math. 46 (1993), 755–81.CrossRefGoogle Scholar
4Chen, G.-C., Levermore, C. and Liu, T. P.. Hyperbolic conservation laws with stiff relaxation terms and entropy. Comm. Pure Appl. Math. 47 (1994), 787–830.CrossRefGoogle Scholar
5Crasta, G. and Piccoli, B.. Viscosity solutions and uniqueness for systems of inhomogeneous balance laws. Discrete Contin. Dynam. Systems 3 (1997), 477–502.CrossRefGoogle Scholar
6Dafermos, C. M. and Hsiao, L.. Hyperbolic systems of balance laws with in homogeneity and dissipation. Indiana Univ. Math. J. 31 (1982), 471–91.CrossRefGoogle Scholar
7Glimm, J.. Solutions in the large for nonlinear hyperbolic systems of equations. Comm. Pure Appl. Math. 18 (1965), 697–715.CrossRefGoogle Scholar
8Greenberg, J. M. and Hsiao, L.. The Riemann problem for the system ut + σx = 0 and (σ – f((u))t + (σ – μf(u)) = 0. Arch. Rational Mech. Anal. 82 (1983), 87–108.CrossRefGoogle Scholar
9Hanouzet, B. and Natalini, R.. Weakly coupled systems of quasilinear hyperbolic equations. Differential Integral Equations. 9 (1996), 1272–92.CrossRefGoogle Scholar
10Hanouzet, B., Natalini, R. and Tasei, A.. On the Chapman–Jouguet limit for a combustion model. SIAM J. Math. Anal, (in press).Google Scholar
11Hsiao, L.. Quasilinear hyperbolic systems and dissipative mechanisms (Singapore: World Scientific, 1998).CrossRefGoogle Scholar
12Jin, S. and Xin, Z.. The relaxation schemes for systems of conservation laws in arbitrary space dimensions. Comm. Pure Appl. Math. 48 (1995), 235–76.CrossRefGoogle Scholar
13Lax, P. D.. Shock waves and entropy. In Contributions to nonlinear functional analysis, ed. Zarantonello, E., 603–34 (New York: Academic Press, 1971).CrossRefGoogle Scholar
14Liu, T. P.. Quasilinear hyperbolic systems. Comm. Math. Phys. 68 (1979), 141–72.CrossRefGoogle Scholar
15Liu, T. P.. Hyperbolic conservation laws with relaxation. Comm. Math. Phys. 108 (1987), 153–75.CrossRefGoogle Scholar
16Natalini, R.. Convergence to equilibrium for the relaxation approximation of conservation laws. Comm. Pure Appl. Math. 49 (1996), 795–823.3.0.CO;2-3>CrossRefGoogle Scholar
17Natalini, R.. A discrete kinetic approximation of entropy solutions to multidimensional scalar conservation laws (Preprint IAC, Quaderno n.9, 1996).Google Scholar
18Schroll, H. J., Tveito, A. and Winther, R.. An error bound for a finite difference scheme applied to a stiff system of conservation laws. SIAM J. Numer. Anal. 34 (1997), 1152–66.CrossRefGoogle Scholar
19Serre, D.. Systémes de his de conservation I (Paris: Diderot, 1996).Google Scholar
20Smoller, J.. Shock waves and reaction–diffusion systems (New York: Springer, 1983).CrossRefGoogle Scholar
21Tartar, L.. Compensated compactness and applications to partial differential equations. In Research notes in Mathematics. Nonlinear Analysis and Mechanics: Heriot–Watt Symposium, Vol. 4, ed. Knops, R. J., 136–212 (New York: Pitman, 1979).Google Scholar
22Tveito, A. and Winther, R.. On the rate of convergence to equilibrium for a system of conservation laws including a relaxation term. SIAM J. Math. Anal. 28 (1997), 136–61.CrossRefGoogle Scholar
23Whitham, J.. Linear and nonlinear waves (New York: Wiley, 1974).Google Scholar