Existence of global-in-time weak solutions for a solidification model with convection in the liquid and rigid motion in the solid. (English) Zbl 1455.35189
Summary: 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.
MSC:
| 35Q35 | PDEs in connection with fluid mechanics |
| 35Q79 | PDEs in connection with classical thermodynamics and heat transfer |
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 35K51 | Initial-boundary value problems for second-order parabolic systems |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35D30 | Weak solutions to PDEs |
| 76A15 | Liquid crystals |
| 76D03 | Existence, uniqueness, and regularity theory for incompressible viscous fluids |
| 80A22 | Stefan problems, phase changes, etc. |
Keywords:
latent heat; interface problem; Allen-Cahn phase-field model; Navier-Stokes-Boussinesq; existence of weak solutionsReferences:
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