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Relative energy approach to a diffuse interface model of a compressible two‐phase flow
Author(s) -
Feireisl Eduard,
Petcu Madalina,
Pražák Dalibor
Publication year - 2019
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.5436
Subject(s) - uniqueness , mathematics , compressibility , limit (mathematics) , compressible flow , mach number , flow (mathematics) , corollary , mathematical analysis , incompressible flow , weak solution , mechanics , physics , geometry , pure mathematics
We propose a simple model for a two‐phase flow with a diffuse interface. The model couples the compressible Navier‐Stokes system governing the evolution of the fluid density and the velocity field with the Allen‐Cahn equation for the order parameter. We show that the model is thermodynamically consistent, in particular, a variant of the relative energy inequality holds. As a corollary, we show the weak‐strong uniqueness principle, meaning any weak solution coincides with the strong solution emanating from the same initial data on the life span of the latter. Such a result plays a crucial role in the analysis of the associated numerical schemes. Finally, we perform the low Mach number limit obtaining the standard incompressible model.