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Global resolvability for a viscoelastic model with relaxation*

Published online by Cambridge University Press:  14 November 2011

Zhu Changjiang
Zhu Changjiang
Affiliation:
Young Scientist Laboratory of Mathematical Physics, Institute of Mathematical Sciences, Academia Sinica, Wuhan 430071, P.R. China
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Abstract

In this paper, we prove the existence of the global smooth solution for the equation of a viscoelastic model with relaxation in time under the only assumption that the C0-norm of the initial data is small, without smallness hypothesis for the C1-norm.

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 125 , Issue 6 , 1995 , pp. 1277 - 1285
Copyright
Copyright © Royal Society of Edinburgh 1995

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References

1Caflisch, R. and Liu, T. P.. Nonlinear stability of shock waves for the Broadwell model. Comm. Math. Phys. 114(1988), 103–30.CrossRefGoogle Scholar
2Chen, G. Q. and Liu, T. P.. Zero relaxation and dissipation limits for hyperbolic conservation laws (IMA Preprint Series 866, Institute of Mathematics and its Applications, Minneapolis, August, 1991).Google Scholar
3Cox, S. E. and Liu, T. P.. Nonlinear stability of traveling waves for a model of viscoelastic materials with memory (Preprint).Google Scholar
4Douglis, A.. Existence theorems for hyperbolic system. Comm. Pure Appl. Math. 5 (1952), 119–54.CrossRefGoogle Scholar
5Greenberg, J. M. and Hsiao, L.. The Riemann problem for system u 1 + σx = 0 and (σ - f(u))t + (σ - μf(u)) = 0. Arch. Rational Mech. Anal. 82 (1983), 87–108.CrossRefGoogle Scholar
6Hoff, D.. Global smooth solutions to quasilinear hyperbolic systems in diagonal form. J. Math. Anal. Appl. 86 (1982), 221–38.Google Scholar
7Hsiao, L. and Tatsien, Li. Global smooth solution of Cauchy problems for a class of quasilinear hyperbolic systems. Chinese Ann. Math. Ser. B 4(1) (1983), 109–15.Google Scholar
8John, F.. Formation of singularities in one-dimensional nonlinear wave propagation. Comm. Pure Appl. Math. 27 (1974), 377–405.CrossRefGoogle Scholar
9Johnson, J. L.. Global continuous solutions of hyperbolic systems of quasilinear equations. Bull. Amer. Math. Soc. 73 (1967), 639–41.Google Scholar
10Johnson, J. L. and Smoller, J. A.. Global solutions for an extended class of hyperbolic systems of conservation laws. Arch. Rational Mech. Anal. 32 (1969), 169–89.Google Scholar
11Lax, P. D.. Development of singularities of solutions of nonlinear hyperbolic partial differential equations. J. Math. Phys. 5 (1964), 611–13.CrossRefGoogle Scholar
12Caizhong, Li and Zhen, Wang. Global smooth solutions for a model equation in viscoelasticity (Preprint).Google Scholar
13Longwei, Lin. Existence of globally continuous for reducible quasilinear hyperbolic systems. Acta Sci. Natur. Univ. Jilin 4 (1963), 83–96.Google Scholar
14Liu, T.-P.. Development of singularities in the nonlinear waves for quasilinear hyperbolic partial differential equations. J. Differential Equations 33 (1979), 92–111.CrossRefGoogle Scholar
15Liu, T.-P.. Hyperbolic conservation laws with relaxation. Comm. Math. Phys. 108 (1987), 153–75.CrossRefGoogle Scholar
16Nishida, T.. Nonlinear hyperbolic equations and related topics in fluid dynamics, Publications Mathematiques d'Orsay 78.02 (Orsay: Dept. de Math., Paris-Sud, 1978).Google Scholar
17Slemrod, M.. Instability of steady shearing flows in a nonlinear viscoelastic fluid. Arch. Rational Mech. Anal. 68 (1978), 211–25.Google Scholar
18Changjiang, Zhu, Caizhong, Li and Huijiang, Zhao. The existence of the global continuous solutions for a class of nonstrict hyperbolic equations. Acta Math. Sci. 14(1) (1994), 96–106.Google Scholar