SIAM Journal on Mathematical Analysis, Volume 52, Issue 6, Page 6260-6280, January 2020.
We introduce a PDE problem modeling a solidification/melting process in bounded two- or three-dimensional domains, coupling a phase-field equation and a Navier--Stokes--Boussinesq system, where the latent heat effect is considered via a modification of the Caginalp model. Moreover, the convection in the nonsolid and solid regions is treated via a phase-dependent viscosity of the material that degenerates in the solid phase, letting only rigid motions in this phase. Then we prove the existence of global-in-time weak solutions (of a regularized problem in three-dimensional domains) based on a limit process of a sequence of dissipative problems furnished truncating the viscosity.

中文翻译：

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具有液体对流和固体刚性运动的凝固模型的全局及时弱解的存在

SIAM数学分析杂志，第52卷，第6期，第6260-6280页，2020年1月。
我们引入了PDE问题，对有界二维或三维域中的凝固/熔化过程进行建模，并耦合了相场方程和Navier。 --Stokes--Boussinesq系统，其中通过修改Caginalp模型来考虑潜热效应。此外，非固相和固相区域中的对流通过在固相中退化的材料的相变粘度来处理，该相中仅允许刚性运动。然后，我们基于一系列耗散性问题的极限过程（证明了截断粘度）证明了实时全局弱解（三维域中正则化问题）的存在。