Ostwald ripening in two dimensions -- the rigorous derivation of the equations from the Mullins-Sekerka dynamics
The authors consider Mullins-Sekerka problem \[ -\Delta\;u=0\quad \text{in }\Omega\setminus\Gamma(t),\quad \frac{\partial u}{\partial \nu}=0\quad\text{on }\partial\Omega,\tag{1} \] \[ u=H,\quad V=\left[\left[\frac{\partial u}{\partial n}\right]\right]\quad \text{on }\Gamma(t), \] where \(H\) is the mean curvature of \(\Gamma(t)\), \(\nu\) is the outward normal to \(\partial \Omega\), \(V\) is the normal velocity of free boundary, \(\left[\left[\frac{\partial u}{\partial n}\right]\right]\) is the jump of derivative to \(\Gamma(t)\). Under the hypothesis that \(\Gamma(0)=\cup_{n=1}^{N}\Gamma_i(0)\) and ``a particle'' \(\Gamma_i(0)\) is a small perturbation of a circle with a small radius, the authors prove the existence of global weak and classical solution to (1) and investigate asymptotic behaviour of particles.
existence of global weak and classical solution
scientific article; zbMATH DE number 2117236