Approximation of Classical Two-Phase Flows of Viscous Incompressible Fluids by a Navier–Stokes/Allen–Cahn System

We show convergence of the Navier–Stokes/Allen–Cahn system to a classical sharp interface model for the two-phase flow of two viscous incompressible fluids with same viscosities in a smooth bounded domain in two and three space dimensions as long as a smooth solution of the limit system exists. Moreover, we obtain error estimates with the aid of a relative entropy method. Our results hold provided that the mobility \(m_\varepsilon >0\) $m_{\epsilon }>0$ in the Allen–Cahn equation tends to zero in a subcritical way, i.e., \(m_\varepsilon = m_0 \varepsilon ^\beta \) $m_{\epsilon }=m_0{\epsilon }^{\beta }$ for some \(\beta \in (0,2)\) $\beta \in (0,2)$ and \(m_0>0\) $m_0>0$ . The proof proceeds by showing via a relative entropy argument that the solution to the Navier–Stokes/Allen–Cahn system remains close to the solution of a perturbed version of the two-phase flow problem, augmented by an extra mean curvature flow term \(m_\varepsilon H_{\Gamma _t}\) $m_{\epsilon }{H}_{{\mathrm{\Gamma }}_t}$ in the interface motion. In a second step, it is easy to see that the solution to the perturbed problem is close to the original two-phase flow.