Abstract
In this paper, we shall prove the existence of solutions for the system of second order partial differential equations. This system is constructed by the phase field equations with a convection described by the Navier-Stokes equations in a liquid region. In our setting, this liquid region is also unknown, which is defined by the solution of the phase field equations. In order to determine the liquid region by the unknown parameter, which is called order parameter, we need to get the continuity. From the L 2 framework, we shall obtain the smoothness of the order parameter by the compactness theorem of Aubin’s type.
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Fukao, T. (2006). Embedding Theorem for Phase Field Equation with Convection. In: Figueiredo, I.N., Rodrigues, J.F., Santos, L. (eds) Free Boundary Problems. International Series of Numerical Mathematics, vol 154. Birkhäuser, Basel. https://doi.org/10.1007/978-3-7643-7719-9_17
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DOI: https://doi.org/10.1007/978-3-7643-7719-9_17
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