Weak solutions to the stationary Cahn-Hillard/Navier-Stokes equations for compressible fluids

We are concerned with the Cahn-Hilliard/Navier-Stokes equations for the stationary compressible flows in a three-dimensional bounded domain. The governing equations consist of the stationary Navier-Stokes equations describing the compressible fluid flows and the stationary Cahn-Hilliard type diffuse equation for the mass concentration difference. We prove the existence of weak solutions when the adiabatic exponent $γ$ satisfies $γ>\frac{4}{3}$. The proof is based on the weighted total energy estimates and the new techniques developed to overcome the difficulties from the capillary stress.

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