On two-phase Stefan problem arising from a microwave heating process
Let \(\Omega\) be a bounded domain in \(\mathbb R^3\) with \(C^1\) boundary \(\partial\Omega\) and consider Maxwell's equations \[ \begin{aligned}\varepsilon \mathbf{E}_t+\sigma\mathbf{E}=\nabla\times\mathbf{H},&\quad (x,t)\in Q_T,\\ \mu\mathbf{H}_t+\nabla \times \mathbf{E}=0, &\quad (x,t)\in Q_T \end{aligned} \] where \(Q_T = \Omega \times (0,T]\). The coefficients in the equation may depend on \(x\) and also on the temperature \(u\), which is one more scalar unknown satisfying the equation \[ A(u)_t-\nabla [k(x,u)\nabla u]=\sigma(x,u)| \mathbf{E}| ^2,\quad (x,t)\in Q_T \] where \(A(u)\) is the graph describing enthalpy, and \(k\) -- as well as \(\sigma\) -- is allowed to depend on the phase. For the thermal problem the no-flux boundary condition is prescribed, while the electric field has to satisfy \[ \mathbf{N}\times\mathbf{E} (x,t)=\mathbf{N}\times\mathbf{G} (x,t),\quad (x,t)\in S_T \] with \(\mathbf{G}\) given on \(S_T=\partial\Omega\times(0,T]\) and \(\mathbf{N}\) is the outward normal to \(\partial \Omega\). The initial values of \(\mathbf{E}, \mathbf{H}, u\) are prescribed. Under some mild assumptions on the data and the coefficients a global existence theorem is proved with reference to a weak formulation. The most restrictive assumption seems to be that \(\sigma\) is bounded. The authors use the classical technique of regularization and compactness. A global existence theorem is provided also for the case of time-harmonic electric and magnetic fields. Finally, it is shown that in the one-dimensional case \(\sigma\) is allowed to have a linear growth with \(u\).
scientific article; zbMATH DE number 5141125