Zero relaxation limit for piecewise smooth solutions to a rate-type viscoelastic system in the presence of shocks. (English) Zbl 0972.35067
Authors’ abstract: We study a rate-type viscoelastic system proposed in I. Suliciu [Int. J. Eng. Sci. 28, No. 8, 829-841 (1990; Zbl 0738.73007)], which is a \(3\times 3\) hyperbolic system with relaxation. As the relaxation time tends to zero, this system converges to the well-known \(p\)-system formally. In the case that the solutions of the \(p\)-system are piecewise smooth, including finitely many noninteracting shock waves, we show that there exist smooth solutions for Suliciu’s model which converge to those of the \(p\)-system strongly as the relaxation time goes to zero. The method used here is the so-called matched asymptotic analysis suggested in J. Goodman and Z. Xin [Arch. Ration. Mech. Anal. 121, No. 3, 235-265 (1992; Zbl 0792.35115)], which includes two parts: the matched asymptotic expansion and stability analysis.
Reviewer: Song Jiang (Beijing)
MSC:
| 35L60 | First-order nonlinear hyperbolic equations |
| 35B25 | Singular perturbations in context of PDEs |
| 35L65 | Hyperbolic conservation laws |
| 35L67 | Shocks and singularities for hyperbolic equations |
| 35L45 | Initial value problems for first-order hyperbolic systems |
| 74D99 | Materials of strain-rate type and history type, other materials with memory (including elastic materials with viscous damping, various viscoelastic materials) |
References:
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