Classical solutions for diffusion-induced grain-boundary motion
The authors consider the evolution of hypersurfaces determined by the mean curvature and by the diffusion of a solute along the surface. As a free boundary problem this consists of a coupled nonlinear partial differential equation. The first sets the normal velocity of the hypersurface, \(\Gamma (t)\), as a function of both the mean curvature (which is the usual model for motion by surface tension) and a deposition effect; while the second is essentially the diffusion of the solute on the manifold: \[ V=-H_{\Gamma}-f(u),\quad\Gamma (0)={\Gamma}_0, \] \[ du/dt={\Delta}_{\Gamma} u - V H_{\Gamma} u + Vu +g(u),\quad\;u(0)=u_0 \] where \(f,g \in C^{\infty}({\mathbb R}, {\mathbb R}), f(0)=0, g(0)=U.\) Here \(du/dt\) is the derivative of the concentration \(u\) of the solute along the flow lines orthogonal to \(\Gamma (t).\) To study this system, the authors represent the moving hypersurface \(\Gamma (t)\) as a graph over a fixed reference manifold, \(\Sigma,\) and reformulate the problem as an evolution over \(\Sigma.\) For the initial data \({\Gamma}_0 \in C^{2+ \beta}, u_0 \in C^{2+ \beta} ({\Gamma}_0)\), they prove the existence, and uniqueness (in a certain class), of a classical solution \((\Gamma (t), u(t))\) on \([0, T)\) for some \(T>0\).
scientific article; zbMATH DE number 1334731