Solutions of an advected phase field system with low regularity velocity

Authors:
Bianca Morelli Rodolfo Calsavara and José Luiz Boldrini

Journal:
Proc. Amer. Math. Soc. **141** (2013), 943-958

MSC (2010):
Primary 35K40, 80A22; Secondary 47H10, 76R10

DOI:
https://doi.org/10.1090/S0002-9939-2012-11467-4

Published electronically:
July 24, 2012

MathSciNet review:
3003687

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Abstract | References | Similar Articles | Additional Information

Abstract: We present a result on existence of solutions for a system of highly nonlinear partial differential equations related to a phase field model for non-isothermal solidification/melting processes in the case of two possible crystallization states and flow of the molten material.

The flow is incompressible with a velocity which is assumed to be given, but with low regularity. We prove the existence of solutions for the associated system and also give estimates for the temperature and the phase fields related to each of the crystallization states in terms of the low regularity norms of the given flow velocity.

These results constitute a fundamental step in the proof of the existence of solutions of a complete model for solidification obtained by coupling the present equations with a singular Navier-Stokes system for the flow velocity. The analysis of this complete model is done in a forthcoming article.

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Additional Information

**Bianca Morelli Rodolfo Calsavara**

Affiliation:
University of Campinas – UNICAMP, School of Applied Sciences, Pedro Zaccaria Street, 1300, CEP 13484-350, Limeira, SP, Brazil

Email:
biancamrc@yahoo.com

**José Luiz Boldrini**

Affiliation:
University of Campinas – UNICAMP, IMECC, Sergio Buarque de Holanda Street, 651, CEP 13083-859, Campinas, SP, Brazil

Email:
bodrini@ime.unicamp.br

DOI:
https://doi.org/10.1090/S0002-9939-2012-11467-4

Keywords:
Parabolic partial differential equations,
fixed point,
solidification,
phase field,
advection,
convection.

Received by editor(s):
July 27, 2011

Published electronically:
July 24, 2012

Additional Notes:
The first author was supported in part by FAPESP, Brazil, Grant 2010/10087-8

The second author was supported in part by CNPq, Brazil, Grant 307833/2008-9

Communicated by:
Walter Craig

Article copyright:
© Copyright 2012
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.