### 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 L_{1}. 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 L^{1}-error is established.

Original language | English (US) |
---|---|

Pages (from-to) | 1115-1135 |

Number of pages | 21 |

Journal | SIAM Journal on Mathematical Analysis |

Volume | 30 |

Issue number | 5 |

DOIs | |

State | Published - Jan 1 1999 |

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### All Science Journal Classification (ASJC) codes

- Analysis
- Computational Mathematics
- Applied Mathematics

### Cite this

*SIAM Journal on Mathematical Analysis*,

*30*(5), 1115-1135. https://doi.org/10.1137/S003614109731984X

}

*SIAM Journal on Mathematical Analysis*, vol. 30, no. 5, pp. 1115-1135. https://doi.org/10.1137/S003614109731984X

**On the zero relaxation limit for a system modeling the motions of a viscoelastic solid.** / Shen, Wen; Tveito, Aslak; Winther, Ragnar.

Research output: Contribution to journal › Article

TY - JOUR

T1 - On the zero relaxation limit for a system modeling the motions of a viscoelastic solid

AU - Shen, Wen

AU - Tveito, Aslak

AU - Winther, Ragnar

PY - 1999/1/1

Y1 - 1999/1/1

N2 - 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.

AB - 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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U2 - 10.1137/S003614109731984X

DO - 10.1137/S003614109731984X

M3 - Article

AN - SCOPUS:0033442490

VL - 30

SP - 1115

EP - 1135

JO - SIAM Journal on Mathematical Analysis

JF - SIAM Journal on Mathematical Analysis

SN - 0036-1410

IS - 5

ER -