The Sharp Interface Limit of the Two-Dimensional Navier--Stokes/Allen--Cahn System as a Brakke Flow

We study the sharp-interface limit of the two-dimensional incompressible Navier--Stokes/Allen--Cahn system with constant mobility and prove the existence of global integral varifold solutions to the coupled Navier--Stokes/mean curvature flow system. The limiting interface measure satisfies Brakke's inequality, and its normal velocity equals the sum of its mean curvature and the normal projection of the fluid velocity.