Uniform attractors for a phase transition model coupling momentum balance and phase dynamics (Q950618)

The long time behaviour of the solutions of nonautonomous dynamical systems related to Frémond's model is studied considered initial-boundary value problem is the following: \[ k\binom{\chi_1}{\chi_2}_t- \eta \binom{\Delta\chi_1}{\Delta\chi_2}+ \binom{{\ell\over\vartheta^*}(\vartheta- \vartheta^*}{\alpha(\vartheta)\operatorname{div}{\mathbf u}}+ \partial I_{{\mathcal K}}(\chi_1, \chi_2)\ni\binom 0 0, \] \[ {\mathbf u}_{tt}+ c{\mathbf u}_t- \operatorname{div} ((-\nu\Delta(\operatorname{div}{\mathbf u})+ \lambda\operatorname{div}{\mathbf u})\mathbb I+ 2\mu\varepsilon({\mathbf u})+ \alpha(\vartheta) \chi_2\mathbb I)= G, \] \[ \chi_1(\cdot, 0)= \chi^0_1,\quad \chi_2(\cdot, 0)= \chi^0_2,\quad{\mathbf u}(\cdot, 0)={\mathbf u}_0,\quad{\mathbf u}_t(\cdot, 0)={\mathbf v}_0\quad\text{in }\Omega, \] \[ \begin{aligned}\partial_n\chi_j= 0&\quad\text{on }\partial\Omega\times (0,+\infty),\quad j= 1,2,\\ {\mathbf u}= \mathbf{0}&\quad\text{on }\partial\Omega\times (0,+\infty),\\ \partial_n(\nu\operatorname{div}{\mathbf u}= 0&\quad\text{on }\partial\Omega\times (0,+\infty). \end{aligned} \] Here, \(k\), \(\eta\), \(\ell\), \(\vartheta^*\), \(c\), \(\lambda\), \(\mu\) are positive parameters and \(\nu\geq 0\). The unknown functions \(\chi_1\), \(\chi_2\) and \({\mathbf u}\) are considered in space-time domain \(Q= \Omega\times (0,+\infty)\), where \(\Omega\subset\mathbb{R}^3\) is fixed bounded regular set. This system arises in the study of the behaviour of a viscoelastic shape memory body subjected to mechanical deformations when the temperature field is prescribed. First, the solution operator is shown to be a semiprocess which is continuous on the proper phase space and satisfies a dissipativity property. Finally, the existence of a unique compact and connected uniform attractor for the system is proved. The crucial step in proving the existence of the uniform attractor relies on the proof of some form of compactness (uniform asymptotic compactness) for the solution operator.