Long-time convergence of solutions to a phase-field system (Q2725351)

The authors consider the Caginalp-Fix phase-field model, NEWLINE\[NEWLINE \partial_t \chi -\Delta \chi + W'(\chi)= \lambda'(\chi)\theta, \qquad\partial_t (\theta + \lambda (\chi))-\Delta \theta =0. NEWLINE\]NEWLINE Here \((t,x) \in \mathbb{R}^+ \times \Omega\), where \(\Omega \in \mathbb{R}^3\) is bounded, \(\chi\) is the phase variable, and \(\theta\) is the temperature. \(\lambda(\cdot)\) is the latent heat, and \(W(\cdot)\) is the two-well energy. Dirichlet conditions are assumed on both \(\chi\) and \(\theta\), though the results of the paper also apply to the more physically relevant case of Neumann boundary conditions on the phase variable \(\chi\). NEWLINENEWLINENEWLINEThe main result of the paper (Theorem 2.1) concerns convergence of solutions to equilibria in the non-generic case when there is a continuum of such equilibria: For any classical solution \(\chi(t,x)\), \(\theta(t,x)\) of the phase-field model above, we have that as \(t \rightarrow \infty\), \(\chi(t,x) \rightarrow \chi_\infty\), \(\theta(t,x) \rightarrow 0\) in \(C(\overline{\Omega})\), where \(\chi_\infty\) is an equilibrium solution. NEWLINENEWLINENEWLINEThe proof uses ideas of Simon which in turn utilize the work of Lojasiewicz on analytic functions of several complex variable. This allows the authors to derive bounds on the functional \(I(\chi)=1/2 \int_\Omega |\nabla \chi|^2 +2 \Gamma (\chi) dx\), where \(\Gamma\) is the primitive of a function obtained by suitably modifying \(W'(\cdot)\). Hence follow bounds on \(\partial_t \chi\) which make it possible to conclude convergence of a solution to a particular element in the set of equilibria.