

Steklov Mathematical Institute Seminar
December 13, 2001, Moscow, Steklov Mathematical Institute of RAS, Conference Hall (8 Gubkina)






On the nonuniqueness of solutions of the equations of the nonlinear theory of elasticity
A. G. Kulikovskii^{} 
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Abstract:
The system of equations of the nonlinear theory of elasticity, which is a hyperbolic system expressing the laws of conservation of mass, momentum, and energy, describes both continuous and discontinuous solutions.
Selfsimilar problems: the problem of disintegration of an arbitrary discontinuity and the ‘piston problem’. Nonuniqueness of the solutions of these problems. Passage to the linear limit as the amplitude of the perturbations tends to zero. Metastable shock wave — the cause of nonuniqueness? Its stability with respect to small perturbations.
Viscoelastic media. Structure of shock waves. Numerical experiments in problems with nonviscous selfsimilar asymptotics; under suitable conditions any of the nonviscous solutions can be realized as the asymptotic behaviour. Nonlinear stability of metastable shock waves.
Concluding remarks. Influence of dispersion on the properties of shock waves.

