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

TL;DR

This paper proves the existence and uniqueness of viscosity solutions for the value function of a controlled local Cahn-Hilliard-Navier-Stokes system in two dimensions, using dynamic programming and comparison principles.

Contribution

It establishes the viscosity solution framework for the value function of the control problem involving the local Cahn-Hilliard-Navier-Stokes system, which was not previously addressed.

Findings

Proved the Dynamic Programming Principle for the value function.

Established the value function as the unique viscosity solution.

Applied comparison principle for uniqueness.

Abstract

In this work, we consider the local Cahn-Hilliard-Navier-Stokes equation with regular potential in two dimensional bounded domain. We formulate distributed optimal control problem as the minimization of a suitable cost functional subject to the controlled local Cahn-Hilliard-Navier- Stokes system and define the associated value function. We prove the Dynamic Programming Principle satisfied by the value function. Due to the lack of smoothness properties for the value function, we use the method of viscosity solutions to obtain the corresponding solution of the infinite dimensional Hamilton-Jacobi-Bellman equation. We show that the value function is the unique viscosity solution of the Hamilton-Jacobi-Bellman equation. The uniqueness of the viscosity solution is established via comparison principle.