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In this paper, we consider the almost sure well-posedness of the Cauchyproblem to the Cahn-Hilliard-Navier-Stokes equation with a randomizationinitial data on a torus $\mathbb{T}^3$. First, we prove the local existence anduniqueness of solution. Furthermore, we prove the global existence anduniqueness of solution and give the relative probability estimate under thecondition of small initial data.

Well-posedness for the Cahn-Hilliard-Navier-Stokes equation with random initial data