On the zero relaxation limit for a system modeling the motions of a viscoelastic solid

Wen Shen, Aslak Tveito, Ragnar Winther

Research output: Contribution to journalArticle

6 Citations (Scopus)

Abstract

We consider a simple model of the motions of a viscoelastic solid. The model consists of a two-by-two system of conservation laws including a strong relaxation term. We establish the existence of a BV-solution of this system for any positive value of the relaxation parameter. We also show that this solution is stable with respect to the perturbations of the initial data in L1. By deriving the uniform bounds, with respect to the relaxation parameter, on the total variation of the solution, we obtain the convergence of the solutions of the relaxation system towards the solutions of a scalar conservation law as the relaxation parameter δ goes to zero. Due to the Lip† bound on the solutions of the relaxation system, an estimate on the rate of convergence towards equilibrium is derived. In particular, an script O sign(√δ) bound on the L1-error is established.

Original languageEnglish (US)
Pages (from-to)1115-1135
Number of pages21
JournalSIAM Journal on Mathematical Analysis
Volume30
Issue number5
DOIs
StatePublished - Jan 1 1999

Fingerprint

Relaxation Limit
System Modeling
Motion
Zero
Conservation
Systems of Conservation Laws
Uniform Bound
Scalar Conservation Laws
Total Variation
Rate of Convergence
Perturbation
Term
Model
Estimate

All Science Journal Classification (ASJC) codes

  • Analysis
  • Computational Mathematics
  • Applied Mathematics

Cite this

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abstract = "We consider a simple model of the motions of a viscoelastic solid. The model consists of a two-by-two system of conservation laws including a strong relaxation term. We establish the existence of a BV-solution of this system for any positive value of the relaxation parameter. We also show that this solution is stable with respect to the perturbations of the initial data in L1. By deriving the uniform bounds, with respect to the relaxation parameter, on the total variation of the solution, we obtain the convergence of the solutions of the relaxation system towards the solutions of a scalar conservation law as the relaxation parameter δ goes to zero. Due to the Lip† bound on the solutions of the relaxation system, an estimate on the rate of convergence towards equilibrium is derived. In particular, an script O sign(√δ) bound on the L1-error is established.",
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On the zero relaxation limit for a system modeling the motions of a viscoelastic solid. / Shen, Wen; Tveito, Aslak; Winther, Ragnar.

In: SIAM Journal on Mathematical Analysis, Vol. 30, No. 5, 01.01.1999, p. 1115-1135.

Research output: Contribution to journalArticle

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