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Gradient flows as a selection procedure for equilibria of nonconvex energies. (English) Zbl 1117.35004

Summary: For atomistic material models, global minimization gives the wrong qualitative behavior; a theory of equilibrium solutions needs to be defined in different terms. In this paper, a concept based on gradient flow evolutions, to describe local minimization for simple atomistic models based on the Lennard-Jones potential, is presented. As an application of this technique, it is shown that an atomistic gradient flow evolution converges to a gradient flow of a continuum energy as the spacing between the atoms tends to zero. In addition, the convergence of the resulting equilibria is investigated in the case of elastic deformation and a simple damaged state.

MSC:

35A15 Variational methods applied to PDEs
35B38 Critical points of functionals in context of PDEs (e.g., energy functionals)
35K55 Nonlinear parabolic equations
26B25 Convexity of real functions of several variables, generalizations
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