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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow><mml:msub><mml:mi>m</mml:mi> <mml:mi>ε</mml:mi></mml:msub> <mml:mo>></mml:mo> <mml:mn>0</mml:mn></mml:mrow> </mml:math> in the Allen-Cahn equation tends to zero in a subcritical way, i.e., <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow><mml:msub><mml:mi>m</mml:mi> <mml:mi>ε</mml:mi></mml:msub> <mml:mo>=</mml:mo> <mml:msub><mml:mi>m</mml:mi> <mml:mn>0</mml:mn></mml:msub> <mml:msup><mml:mi>ε</mml:mi> <mml:mi>β</mml:mi></mml:msup> </mml:mrow> </mml:math> for some <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>β</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> <mml:mo>)</mml:mo></mml:mrow> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow><mml:msub><mml:mi>m</mml:mi> <mml:mn>0</mml:mn></mml:msub> <mml:mo>></mml:mo> <mml:mn>0</mml:mn></mml:mrow> </mml:math> . 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow><mml:msub><mml:mi>m</mml:mi> <mml:mi>ε</mml:mi></mml:msub> <mml:msub><mml:mi>H</mml:mi> <mml:msub><mml:mi>Γ</mml:mi> <mml:mi>t</mml:mi></mml:msub> </mml:msub> </mml:mrow> </mml:math> 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.