Exact Solution of Nonlinear Boundary Value Problems for Surface
by Professor Philip Broadbridge
Abstract: Curvature-driven surface diffusion on crystalline surfaces is modelled by the nonlinear 4th order Mullins equation. There is a class of nonlinear weakly anisotropic models that is fully integrable. Exact solutions are constructed for development of a grain boundary groove and for smoothing of an initial ramp dislocation.
Even for linear fourth order "diffusion", there are strange overshoot phenomena that are no longer proscribed by maximum principles of second
There are additional phenomena due entirely to the nonlinearity. For example, in a solvable quasilinear model, the depth of a grain boundary
groove remains bounded as the dihedral angle approaches vertical.
At a dislocation point of infinite curvature, the quasilinear Mullins model should be extended to a fully nonlinear degenerate model to
account for Gibbs-Thompson evaporation-condensation. An exactly solvable fully nonlinear degenerate diffusion model shows that unlike in the
quasilinear model, deposition rate at the dislocation point is bounded, and the slope remains discontinuous for a finite delay time.
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