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Equations with singular diffusivity. (English) Zbl 0952.74014

Summary: We deal with the models of faceted crystal growth and of grain boundaries based on gradient systems with nondifferentiable energy. We study their most basic forms given by the equations \(u_t=(u_x/ |u_x |)_x\) and \(u_t=(1/a) (au_x/ |u_x|)_x\), where both of the related energies include the \(|u_x|\) term of power one which is nondifferentiable at \(u_x=0\). The first equation is spatially homogeneous, while the second one is spatially inhomogeneous when \(a\) depends on \(x\). These equations naturally express nonlocal interactions through their singular diffusivities (infinitely large diffusion constant), which make the profiles of the solutions completely flat. We give the mathematical basis for justifying and analyzing these equations, and develop theoretical and numerical approaches which show how the solutions of the equations evolve.

MSC:

74E15 Crystalline structure
74G65 Energy minimization in equilibrium problems in solid mechanics
35K65 Degenerate parabolic equations
82D25 Statistical mechanics of crystals
74N25 Transformations involving diffusion in solids
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