Cahn-Hilliard-Navier-Stokes system describes the evolution of two isothermal, incompressible, immiscible fluids in a bounded domain. In this work, we consider the stationary nonlocal Cahn-Hilliard-Navier-Stokes system in two and three dimensions with singular potential. We prove the existence of a weak solution for the system using pseudo-monotonicity arguments and Browder's theorem. Further we establish the uniqueness and regularity results for the weak solution of the stationary nonlocal Cahn-Hilliard-Navier-Stokes system for constant mobility parameter and viscosity. Finally, in two dimensions, we establish that the stationary solution is exponentially stable under suitable conditions on mobility parameter and viscosity.

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关于平稳非局部 Cahn-Hilliard-Navier-Stokes 系统：存在性、唯一性和指数稳定性

Cahn-Hilliard-Navier-Stokes 系统描述了有界域中两种等温、不可压缩、不混溶的流体的演化。在这项工作中，我们考虑具有奇异势的二维和三维的平稳非局部 Cahn-Hilliard-Navier-Stokes 系统。我们使用伪单调性参数和布劳德定理证明了系统的弱解的存在。此外，我们建立了固定非局部 Cahn-Hilliard-Navier-Stokes 系统的弱解的唯一性和规律性结果，用于恒定的流动参数和粘度。最后，在二维中，我们确定了在流动参数和粘度的合适条件下，固定解是指数稳定的。