The quasistationary phase field equations with Neumann boundary conditions (Q1566849)

The paper considers the quasistationary phase field equations \[ \partial_t(u+ \varphi)- \Delta u= f\quad\text{in }\Omega\times ]0,T[, \] \[ \partial_\nu u= 0\quad\text{on }\partial\Omega\times ]0, T[, \] \[ (u+\varphi)(0)= w_0, \] and \[ -2\varepsilon\Delta\varphi+ {1\over\varepsilon} W'(\varphi)= u\quad\text{in }\Omega\times ]0, T[, \] \[ \partial_\nu\varphi= 0\quad\text{on }\partial\Omega\times ]0, T[ \] and shows that with a double-well potential \(W(t)= (t^2- 1)^2\) and a space dimension \(n\leq 3\) the system has a solution. Moreover, it is shown that as \(\varepsilon\to 0\) the solutions converge to the solution of the Stefan problem with the Gibbs-Thomson law. The crucial point here is the compactness of \(\varphi\) which is proved not by a compact embedding argument, but by introducing a time discrete approximation.