Well-posedness for a non-isothermal flow of two viscous incompressible fluids

Juliana Honda Lopes; Gabriela Planas

doi:10.3934/cpaa.2018117

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2018, Volume 17, Issue 6: 2455-2477. Doi: 10.3934/cpaa.2018117

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Well-posedness for a non-isothermal flow of two viscous incompressible fluids

Juliana Honda Lopes and
Gabriela Planas^,

Departamento de Matemática, Instituto de Matemática, Estatística e Computação Científica, Universidade Estadual de Campinas, Rua Sergio Buarque de Holanda, 651, 13083-859, Campinas - SP, Brazil

* Corresponding author
* Corresponding author

Received: October 2017

Revised: January 2018

Published: June 2018

G. Planas was partially supported by CNPq-Brazil, grants 306646/2015-3 and 402388/2016-0. J. H. Lopes was supported by Capes-Brazil and CNPq-Brazil, grant 143214/2015-2.

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Abstract

This work is concerned with a non-isothermal diffuse-interface model which describes the motion of a mixture of two viscous incompressible fluids. The model consists of modified Navier-Stokes equations coupled with a phase-field equation given by a convective Allen-Cahn equation, and energy transport equation for the temperature. This model admits a dissipative energy inequality. It is investigated the well-posedness of the problem in the two and three dimensional cases without any restriction on the size of the initial data. Moreover, regular and singular potentials for the phase-field equation are considered.

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Mathematics Subject Classification: Primary: 35Q35, 35A01; Secondary: 35K20.

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