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 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
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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 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 -