Optimal Boundary Control for the Stochastic Allen-Cahn Navier-Stokes system in two dimensions

TL;DR

This paper investigates optimal boundary control for a complex stochastic PDE system coupling Navier-Stokes and Allen-Cahn equations in two dimensions, establishing well-posedness and existence of optimal solutions.

Contribution

It introduces a novel control framework for the stochastic Allen-Cahn Navier-Stokes system and proves existence and uniqueness of solutions in 2D.

Findings

Proved well-posedness of the coupled stochastic system.

Established existence of an optimal boundary control.

Demonstrated the mathematical robustness of the control approach.

Abstract

In this work, we study an optimal boundary control for the stochastic Allen Cahn Navier Stokes system. The governing system of nonlinear partial differential equations consists of the stochastic Navier Stokes equations with non homogeneous Navier slip boundary condition coupled with a phase-field equation, which is the convective Allen Cahn equation type. We investigate the well posedness of the nonlinear system. More precisely, the existence and uniqueness of global strong solution in dimension two is established and we prove the existence of an optimal solution to the control problem.