Phase-field models with hysteresis
The authors consider the system of equations \(\mu(\theta)w_t+f_1[w]+\theta f_2[w]=0\), \((\theta+F_1[w])_t-\Delta\theta=\Psi(x,t,\theta)\) with \(x\in\Omega\subset\mathbb{R}^N\), where \(\Omega\) is a bounded domain with Lipschitz boundary, \(t\in(0,T)\), and \(f_1, f_2, F_1\) are hysteresis operators. In the context of the main application, i.e. phase-field equations modelling the phase transitions, \(\theta\) is the absolute temperature, \(\mu\) the phase relaxation coefficients, \(\Psi\) is a heat source, and the definition of hysteresis operators is linked to the definition of free energy. The system is supplemented by initial and boundary conditions \(\theta(x,0)=\theta^0(x)\), \(w(x,0)=w^0(x)\) in \(\Omega\), and \(\frac{\partial\theta}{\partial n}=0\) on \(\partial\Omega\times(0,T)\). The main result is an existence and uniqueness theorem, under suitable assumptions on \(\mu, f_1, f_2, F_1, \Psi\), and on the data \(\theta^0, w^0\). The solution satisfies the above system of equations a.e., and additionally an estimate of the type \(\theta \geq \delta e^{-\beta t}\) is proved. The technical part of the paper is preceded by a useful introduction in which the authors clearly illustrate the physical relevance of the problem. The paper extends in a significant way previous results by the same authors.
scientific article; zbMATH DE number 1563456