On singular Navier-Stokes equations and irreversible phase transitions

José Luiz Boldrini; Luís H. de Miranda; Gabriela Planas

doi:10.3934/cpaa.2012.11.2055

Article Contents

2012, Volume 11, Issue 5: 2055-2078. Doi: 10.3934/cpaa.2012.11.2055

This issue Previous Article Smooth attractors for the Brinkman-Forchheimer equations with fast growing nonlinearities Next Article An abstract existence theorem for parabolic systems

On singular Navier-Stokes equations and irreversible phase transitions

1.
Departamento de Matemática, Instituto de Matem
2.
Departamento de Matem
3.
Departamento de Matemática, IMECC - UNICAMP, Rua Sergio Buarque de Holanda, 651, 13083-859 Campinas, SP

Received: April 2010

Revised: December 2011

Published: September 2012

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Abstract

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.

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Mathematics Subject Classification: Primary: 35K55, 35K67, 35Q30, 35R35; Secondary: 47H05, 80A22.

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