On Cahn-Hilliard-Navier-Stokes equations with Nonhomogeneous Boundary

TL;DR

This paper investigates the well-posedness of the Cahn-Hilliard-Navier-Stokes system with nonhomogeneous boundary conditions, establishing existence, uniqueness, and convergence of solutions in two-dimensional bounded domains.

Contribution

It provides new results on existence, uniqueness, and long-term behavior of solutions for the CHNS system with nonhomogeneous boundary conditions in 2D.

Findings

Existence of global weak solutions in 2D.

Uniqueness of weak solutions via continuous dependence.

Convergence of solutions to stationary states.

Abstract

The evolution of two isothermal, incompressible, immiscible fluids in a bounded domain is governed by Cahn-Hilliard-Navier-Stokes equations (CHNS System). In this work, we study the well-posedness results for the CHNS system with nonhomogeneous boundary condition for the velocity equation. We obtain the existence of global weak solutions in the two-dimensional bounded domain. We further prove the continuous dependence of the solution on initial conditions and boundary data that will provide the uniqueness of the weak solution. The existence of strong solutions is also established in this work. Furthermore, we show that in the two-dimensional case, each global weak solution converges to a stationary solution.