Abstract
We consider a model of a two–phase flow based on the phase field approach, where the fluid bulk velocity obeys the standard Navier–Stokes system while the concentration difference of the two fluids plays a role of order parameter governed by the Allen–Cahn equations. Possible thermal fluctuations are incorporated through a random forcing term in the Allen–Cahn equation. We show that suitable dissipative martingale solutions satisfy a stochastic version of the relative energy inequality. This fact is used for showing the weak–strong uniqueness principle both pathwise and in law.
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