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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.

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