# zbMATH — the first resource for mathematics

Blow-up in nonlinear three-dimensional thermoelasticity. (English) Zbl 0705.35081
It is shown that solutions of the equations of nonlinear thermoelasticity in general will develop singularities in finite time: There are singularities for which no global $$C^ 2$$-plane wave solution exists. The idea is to transfer the classical linear decomposition of the displacement vector into a curl-free and a divergence-free part resp. to the nonlinear equation.
Reviewer: R.Racke

##### MSC:
 35L67 Shocks and singularities for hyperbolic equations 74B20 Nonlinear elasticity 35Q72 Other PDE from mechanics (MSC2000)
Full Text:
##### References:
 [1] Christodoulou, Comm. Pure Appl. Math. 39 pp 267– (1986) [2] Fujita, J, Fac. Sci. Univ. Tokyo, Sect. I 13 pp 109– (1966) [3] ’Global existence and asymptotic behavior of smooth solutions in one-dimensional nonlinear thermoelasticity’, Thesis, University of Bonn, 1988. [4] John, Comm. Pure Appl. Math. 27 pp 377– (1974) [5] John, Comm. Pure Appl. Math. 34 pp 29– (1981) [6] ’Formation of singularities in elastic waves’, in Ciarlet, P. G. and Roseau. M. (eds) Trends and Applications of Pure Mathematics to mechanics, Proceedings Palaiseau 1983, Lecture Notes in Physics 195, 1984, pp. 194-210. [7] John, Comm. Pure Appl. Math. 41 pp 615– (1988) [8] ’Systems of a hyperbolic-parabolic composite type, with applications to the equations of magnetohydrodynamics’, Thesis, Kyoto University, 1983. [9] Klainerman, Arch. Rat. Mech. Anal. 78 pp 73– (1982) [10] Klainerman, Comm. Pure Appl. Math. 38 pp 321– (1985) [11] Klainerman, Lec. Appl. Math. 23 pp 293– (1986) [12] Liu, J. Differential Equations 33 pp 92– (1979) [13] Ponce, Nonlinear Anal. T. M. A. 9 pp 399– (1985) [14] Racke, Math. Z. [15] Slemrod, Arch. Rat. Mech. Anal. 76 pp 97– (1981) [16] Zheng, Scienta Sinica (Ser. A) 30 pp 1122– (1987)
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.