Global Weak Solutions to an Initial-Boundary Value Problem for a Three-phase Field Model of Solidification

Abstract

In this article, we study an initial-boundary value problem for a three-phase field model of nonisothermal solidification processes in the case of two possible crystallization states. The governing equations of the model are the three phase-field equations coupled with a nonlinear heat equation. Each equation of the model has strong nonlinearities involving the higher-order derivatives. We prove the existence of global-in-time weak solutions to our problem for one-dimensional case.

Weak Solutions to an Initial-Boundary Value Problem of a Consistent Three-phase-field Model

“…Second, rewriting the model by moving the second term of the first three equations of (8) to the right, we arrive at Similarly, we calculate the following estimates Applying the energy method similarly to deal with (15), we calculate the estimates of higher order derivation. Lemma 6 Under the hypothesis (10) . Then Theorem 2 is completely proved.…”

“…Tang [8][9] investigated the well-posedness of global weak solution for a two-phase-field model which does not directly show an expression for temperature and a three-phase-field model with different methods, respectively. While Akram [10] showed an expression for temperature and discussed the global in-time weak solution to a three-phase-field model that describes the evolution of sea ice in one-dimensional case. In their three-phase-field models, there's no spurious phase.…”