On local uniqueness in nonlinear elastodynamics

Abstract

Abstract.A conservation law, derived from properties of the energy-momentum tensor, is used to establish uniqueness of suitably constrained solutions to the initial boundary value problem of nonlinear elastodynamics. It is assumed that the region is star-shaped, that the data are affine, and that the strain-energy function is strictly rank-one convex and quasi-convex. It is shown how these assumptions may be successively relaxed provided that the class of considered solutions is correspondingly further constrained.1. Introduction. This paper examines uniqueness of locally smooth solutions to the initial displacement boundary value problem of nonlinear homogeneous elastodynamics in the absence of body-forces and on a bounded three-dimensional region. Weak solutions are not necessarily unique, and smooth solutions do not in general exist globally in time. Consequently, uniqueness is sought only in the class of sufficiently smooth solutions on the maximal time interval of existence. s is the usual Hilbert space of order s, and s > n/2 + 1. The discussion in [7], that extends to thermoelastodynamics, considers a strongly elliptic entropy function and from estimates for continuous dependence on initial data obtains uniqueness of a smooth solution in the class of weak solutions. A similar result is established in [8] for a poly-convex entropy subject to certain other conditions. A different convexity assumption, further examined by Sofer [25], is adopted by Knops, Levine and Payne [20], who likewise establish uniqueness as a consequence of continuous dependence estimates.