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On singular Navier-Stokes equations and irreversible phase transitions

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  • We analyze a singular system of partial differential equations corresponding to a model for the evolution of an irreversible solidification of certain pure materials by taking into account the effects of fluid flow in the molten regions. The model consists of a system of highly non-linear free-boundary parabolic equations and includes: a heat equation, a doubly nonlinear inclusion for the phase change and Navier-Stokes equations singularly perturbed by a Carman-Kozeny type term to take care of the flow in the mushy region and a Boussinesq term for the buoyancy forces due to thermal differences. Our approach to show existence of generalized solutions of this system involves time-discretization, a suitable regularization procedure and fixed point arguments.
    Mathematics Subject Classification: Primary: 35K55, 35K67, 35Q30, 35R35; Secondary: 47H05, 80A22.


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