Abstract
This paper is devoted to the study of a non-isothermal incompressible Navier–Stokes–Allen–Cahn system which can be considered as a model describing the motion of the mixture of two viscous incompressible fluids. This kind of models is physically relevant for the analysis of non-isothermal fluids. The governing system of nonlinear partial differential equations consists of the Navier–Stokes equations coupled with a phase-field equation, which is the convective Allen–Cahn equation type, and an energy transport equation for the temperature. We investigate the well-posedness of the nonlinear system. More precisely, existence and uniqueness of local strong solutions in two and three dimensions for any initial data are proved. Moreover, existence of global weak solutions and existence and uniqueness of global strong solution in dimension two, when the initial temperature is suitably small, are established.
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Acknowledgements
This work was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior-Brasil (CAPES)-Finance Code 001. GP was partially supported by CNPq-Brazil, grants 308093/2018-6 and 402388/2016-0, and FAPESP-Brazil grant 19/02512-5.
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Communicated by David Lannes.
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Honda Lopes, J., Planas, G. On a non-isothermal incompressible Navier–Stokes–Allen–Cahn system. Monatsh Math 195, 687–715 (2021). https://doi.org/10.1007/s00605-021-01564-2
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DOI: https://doi.org/10.1007/s00605-021-01564-2