zbMATH — the first resource for mathematics

Weak solutions for a nonlinear system in viscoelasticity. (English) Zbl 0635.73047
This paper contains an extension of work by R. J. DiPerna [Arch. Ration. Mech. Anal. 82, 27-70 (1983; Zbl 0519.35054)] on the global existence of \(L^{\infty}\)-weak solutions to the Cauchy problem \(w_ t=v_ x\), \(v_ t=\sigma_ x\), \(x\in {\mathbb{R}}\), \(t>0\). The constitutive equation \(\sigma =\phi (w)\) used previously by DiPerna to model a one- dimensional elastic medium is here replaced by the assumption \(\sigma =\phi (w(x,t))+\int^{t}_{0}k(t-\tau)\phi (w(x,\tau)d\tau\), where the integral term is used to represent a type of fading memory. The solutions (w,v) to the Cauchy problem belong, for arbitary finite \(T>0\), to the class \(L^{\infty}((0,T)\); L 2(\({\mathbb{R}}))\cap L^{\infty}((0,T)\times {\mathbb{R}})\).
Reviewer: R.Saxton

74Hxx Dynamical problems in solid mechanics
35A15 Variational methods applied to PDEs
45K05 Integro-partial differential equations
74C10 Small-strain, rate-dependent theories of plasticity (including theories of viscoplasticity)
74B20 Nonlinear elasticity
35L99 Hyperbolic equations and hyperbolic systems
35L65 Hyperbolic conservation laws
Full Text: DOI
[1] Boldrini J. L., LCDS report 85 (1985)
[2] DOI: 10.1007/BF00250741 · Zbl 0595.73026
[3] Dafermos C. M., Math. Anal 18 pp 409– (1987)
[4] Dafermos C. M., Solutions in L for a conservation law with memory (1986)
[5] DOI: 10.1080/03605307908820094 · Zbl 0464.45009
[6] Dafermos C. M., Arner. J. Math. Supplement 4 pp 87– (1981)
[7] DOI: 10.1007/BF00251724 · Zbl 0519.35054
[8] DOI: 10.1007/BF00251590 · Zbl 0145.46203
[9] DOI: 10.1016/0022-0396(85)90147-0 · Zbl 0535.35057
[10] Lady┼żenskaja O. A., Arner. Math. Soc 23 (1968)
[11] Lax, P. D. 1971.Shock waves and entropy, in Contributions to Nonlinear Functional Analysis, Edited by: Zarantonello, E. A. 603–634. Academic Press.
[12] Maccamy R. C., Quart. Appl. Math 35 pp 21– (1977)
[13] Murat F., Ann. Scuola Norm. Sup. Pisa Set. Fis. Mat 5 pp 489– (1978)
[14] Murat F., J. Math Pures Appi 60 pp 309– (1981)
[15] Nishida T., Publ. Math. Orsay 60 pp 46– (1978)
[16] Nohel J. A., IMA Volumes in Mathematics and its Applications, Springer Lecture Notes in Mathematics,
[17] Rascle M., Lectures in Appl. Math 23 pp 359– (1986)
[18] Renardy M., Mathematical Problems in Viscoelasticity, Longman Group Limited · Zbl 0719.73013
[19] Slemrod M., Arch. Rational Mech. Anal 68 pp 211– (1978)
[20] DOI: 10.1137/0511071 · Zbl 0464.45010
[21] DOI: 10.1007/BFb0061808
[22] Tartar L., Nonlinear Analysis and Mechanics: Heriot-Watt Symposium 4 (1979)
[23] Tartar L., Systems of Nonlinear Partial Differential Equations (1983) · Zbl 0536.35003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.