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In this work, we consider the local Cahn-Hilliard-Navier-Stokes equation withregular potential in two dimensional bounded domain. We formulate distributedoptimal control problem as the minimization of a suitable cost functionalsubject to the controlled local Cahn-Hilliard-Navier- Stokes system and definethe associated value function. We prove the Dynamic Programming Principlesatisfied by the value function. Due to the lack of smoothness properties forthe value function, we use the method of viscosity solutions to obtain thecorresponding solution of the infinite dimensional Hamilton-Jacobi-Bellmanequation. We show that the value function is the unique viscosity solution ofthe Hamilton-Jacobi-Bellman equation. The uniqueness of the viscosity solutionis established via comparison principle.

Existence and Uniqueness of viscosity solutions of Value function of Local Cahn-Hilliard-Navier-Stokes system