Solutions of an advected phase field system with low regularity velocity

Calsavara, Bianca; Boldrini, José

doi:10.1090/S0002-9939-2012-11467-4

Solutions of an advected phase field system with low regularity velocity
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by Bianca Morelli Rodolfo Calsavara and José Luiz Boldrini PDF

Proc. Amer. Math. Soc. 141 (2013), 943-958 Request permission

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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