For the equations of elastodynamics with polyconvex stored energy, and some related simpler systems, we define a notion of dissipative measure-valued solution and show that such a solution agrees with a classical solution with the same initial data when such a classical solution exists. As an application of the method we give a short proof of strong convergence in the continuum limit of a lattice approximation of one dimensional elastodynamics in the presence of a classical solution. Also, for a system of conservation laws endowed with a positive and convex entropy, we show that dissipative measure-valued solutions attain their initial data in a strong sense after time averaging.

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Problems involving percolation in liquids (i.e., involving connectivity of some sort) range from the metal–insulator transition in liquid metals to the properties of supercooled water. A common theme, however, is that connectivity can be distinguished from interaction and that one should not be slighted in order to describe the other. In this paper we suggest a model for percolation in liquids—the model of extended spheres—which permits connectivity to be studied in the context of, but independently from, liquid structure. This model is solved exactly in the Percus–Yevick approximation, revealing the existence of an optimum liquid structure for percolation. We analyze this behavior by first deriving an explicit diagrammatic representation of the Percus–Yevick theory for connectivity and then studying how the various diagrams contribute. The predictions are in excellent qualitative agreement with recent Monte Carlo calculations.

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