The phase field equations and gradient flows of marginal functions (Q2760725)

The problem of the article consists in finding a function \(\vartheta(x,t)\) and a vector-valued function \(\varphi(x,t)=(\varphi_1(x,t),\dots,\varphi_n(x,t))\) in the cylinder \(Q_T=\Omega\times[0,T]\) with lateral surface \(\Gamma=\partial\Omega\times[0,T]\) satisfying the equations: NEWLINE\[NEWLINE {\partial\over\partial t}(\vartheta+\varphi_1)=\Delta \vartheta,\quad -\Delta\varphi+\nabla_\varphi\Phi(\varphi)=\vartheta\vec{e}_1, NEWLINE\]NEWLINE the conditions on \(\Gamma \): NEWLINE\[NEWLINE \frac{\partial\vartheta}{\partial \vec{n}}+\lambda\vartheta=0, \quad\frac{\partial\varphi}{\partial \vec{n}}+\mu\varphi=0, NEWLINE\]NEWLINE and the conditions for \(t=0\): NEWLINE\[NEWLINE \vartheta(x,0)+\varphi_1(x,0)=-u_0(x),\quad x\in\Omega. NEWLINE\]NEWLINE Here \(\vec{n}\) is the outward normal to \(\partial\Omega\), \(\lambda\) and \(\mu\) are constants, while \(\{e_i\}\) are the entries of the canonical basis for \(\mathbb R^d\). The above problem describes the phase transitions of a medium. For this problem a generalized solution is defined as a vector-valued function \((\vartheta,\varphi)\) that satisfies some integral identity so that, for a fixed function \(\vartheta\), the vector-valued function \(\varphi(t)\) must be a critical point of the Ginzburg-Landau free energy functional NEWLINE\[NEWLINE F(\vartheta,\varphi)= \int_\Omega\left[\frac{|\nabla\varphi|^2}{2}+ \Phi(\varphi)-\frac{\vartheta^2}{2}-\vartheta\varphi_1\right]dx+ \frac{\mu}{2}\int_{\partial\Omega}|\varphi|^2 ds. NEWLINE\]NEWLINE Since this functional has many critical points, it is necessary to distinguish appropriate solutions. The authors suggest two physical principles as selection rules: the maximal entropy principle and the minimal entropy production principle. The proof of the existence of an appropriate generalized solution is the main result of this article.