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Global solution to a nonlinear system for irreversible phase changes. (English) Zbl 0956.35122
Summary: The following nonlinear evolution system is investigated \[ \partial_t(\vartheta+\chi)- \Delta\vartheta= 0\quad\text{in }Q:= \Omega\times ]0,T[, \] \[ \Delta\chi+ \alpha(\partial_x\chi)+ \beta(\chi)\ni \vartheta\quad\text{in }Q, \] \[ \vartheta(\cdot,0)= \vartheta_0,\quad \chi(\cdot,0)= \chi_0\quad\text{in }\Omega, \] \[ \partial_n\chi= \partial_n\vartheta= 0\quad\text{in }\partial\Omega\times ]0,T[, \] where \(\Omega\subset \mathbb{R}^3\) is a bounded domain with smooth boundary \(\partial\Omega\), \(\alpha\) and \(\beta\) are two maximal monotone graphs. It describes a wide class of phase transition problems, including irreversible phase changes. The existence of a solution is established through a suitable regularization procedure together with a time discretization. A uniqueness result is also given under some further assumptions.

35Q72 Other PDE from mechanics (MSC2000)
82C26 Dynamic and nonequilibrium phase transitions (general) in statistical mechanics
80A22 Stefan problems, phase changes, etc.