Maximal attractor and inertial sets for a conserved phase field model
It is the concern of the authors to investigate existence and properties of solutions of an equation due to G. Caginalp, which will not be written out. This equation contains two unknowns, the temperature \(u\) and an order parameter \(\varphi\) coming from thermodynamics, \(\varphi\) describes the phase of the underlying substance. The case \(\varphi\) close to 1 corresponds to the liquid phase, \(\varphi\) close to \(- 1\) to the solid phase. The interphase between solid and liquid is described by the set of points where \(\varphi= 0\). This equation is favorably replaced by another one via the substitution \(\varphi= \varphi\), \(v= u+ {\ell\over 2} \varphi\), \(\ell\) a parameter \(> 0\). The resulting system is: \[ \tau\varphi_t= - \xi^2 \Delta(\xi^2 \Delta \varphi- g(\varphi)- \ell\varphi+ 2v),\;v_t= K\Delta v- K\textstyle{{\ell\over2}} \Delta\varphi.\tag{1} \] The constants \(\tau\), \(\xi\), \(K\), \(\ell\) are all \(> 0\) and have physical meaning, and \(g(s)\) is a polynomial subject to some conditions; in the simplest of all cases \(g(s)= {1\over 2}(s^3- s)\). System (1) is considered on \(\Omega\times \mathbb{R}_+\) with \(\Omega\subseteq \mathbb{R}^n\), \(n= 1,2,3\) smooth and bounded. There are Neumann boundary conditions \(\partial_n \varphi= \partial_n \Delta \varphi= \partial_n v= 0\) with \(\partial_n\) the outer unit normal derivative at \(x\in \partial \Omega\). There are initial conditions \(\varphi(x, 0)= \varphi_0(x)\), \(v(x, 0)= v_0(x)\). A functional analytic frame is imposed on (1). There are reasons to consider (1) on the spaces \(H= H^1(\Omega)'\times L^2(\Omega)\) (\(H^1(\Omega)'= \text{dual}\) of \(H^1(\Omega)\)) and \(V\) which consists of the pairs \((\varphi, v)\) in \(H^2(\Omega)\times H^1(\Omega)\) such that \(\partial_n \varphi= 0\). The fact that \(\int_\Omega v(x, t) dx\), \(\int_\Omega \varphi(x, t) dx\) are conserved gives rise to the set \(H_{\beta\gamma}\), \(\widetilde H_{\beta\gamma}\) where \(H_{\beta\gamma}\) is roughly speaking the set of \((\varphi, v)\in H\) with \(\int_\Omega \varphi(x) dx= \beta|\Omega|\), \(\int_\Omega v(x) dx= \gamma|\Omega|\). The authors prove a series of theorems which can only be indicated. The first states among others that if \((\varphi_0, v_0)\in H\) then (1) admits a unique global solution \((\varphi, v)\in H\) such that for any \(T> 0\) we have that \((\varphi, v)\) is in both \(L^\infty(0, T; H_{\beta\gamma})\) and \(L^2(0, T; H^1(\Omega)^3)\). The next theorem provides information about smoothness properties: if \((\varphi_0, v_0)\in L^2(\Omega)^2\) then \((\varphi, v)\) is in \(C([0, \infty), L^2(\Omega)^2)\). Further results about the existence of global attractors and of inertial sets are proved. The basic technical tool from which all the relevant information is extracted is a suitable Galerkin method. The necessary estimates are quite subtle and elaborate.
scientific article; zbMATH DE number 883511