The Cahn-Hilliard equation with dynamic boundary conditions.

*(English)*Zbl 1035.35050The paper deals with the initial and boundary value problem (IBVP) for the Cahn-Hilliard equation
\[
\chi_t = \Delta \mu,\quad \mu = -\Delta \chi -\chi + \chi^3
\]
in a bounded domain \(\Omega\) with the following boundary and initial conditions
\[
\partial_\nu \mu| _{\partial\Omega} =0, \quad \left(\sigma_s \Delta_{| | } \chi -\partial_\nu \chi -g_s \chi +h_s-1/\Gamma_s\, \chi_t\right)| _{\partial\Omega}=0,
\]
\[
\chi(x,0) = \chi_0(x), \quad x\in \Omega.
\]
Here \(\partial_\nu\) denotes the exterior normal derivative, \(\Delta_{| | }\) represents the tangential Laplacian, and \(\sigma_s, g_s\), \(h_s\) and \(\Gamma_s\) are constants. Such an initial and boundary value problem arises in the modeling of decomposition of binary alloys that also interact with the wall in a short range. Assuming \(\chi_0\in H^3\), the authors establish the existence and uniqueness of strong solutions to this IBVP. Mathematically, the major new feature of this problem is that the boundary conditions involve the time-derivative and the tangential Laplacian. To overcome the difficulties associated with this new feature, the authors first deal with an approximation problem (\(P_\epsilon\)) for a small \(\epsilon>0\). After showing the existence and uniqueness of solutions to (\(P_\epsilon\)) and obtaining uniform bounds for these solutions, they then pass to the limit. The limit solves the original IBVP. The uniqueness of the solutions follows from energy estimates. This paper is well written and reading it is a pleasure.

Reviewer: Jiahong Wu (Stillwater)