Optimal control of the phase interface during the solidification of a GaAs melt

G Ω We consider the depicted crucible Ω = G×H , with th solid phase Ωs, the liquid phase Ωl und the free boundary Γ. The solidification process is modelled using the heat equation (1) in the solid phase Ωs and in the liquid phase Ωl. In the liquid phase additionally flow driven by convection and near-wall Lorentz forces (3), (4) are considered, where the convection term is modelled using the Boussinesq approximation. The phase transition is constituted by the Stefan condition (5) and the melting temperature condition (7) at the free boundary Γ. The Robin-type boundary condition (6) models the heat transfer at the crucible wall ∂Ω. Altogether one obtains the nonlinear system