Sharp Interface Limit for a Navier-Stokes/Allen-Cahn System with Different Viscosities

Helmut Abels; Mingwen Fei

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arXiv · 2201.09343

Sharp Interface Limit for a Navier-Stokes/Allen-Cahn System with Different Viscosities

Helmut Abels·Mingwen Fei

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Abstract

We discuss the sharp interface limit of a coupled Navier-Stokes/Allen-Cahn system in a two dimensional, bounded and smooth domain, when a parameter $\varepsilon>0$ that is proportional to the thickness of the diffuse interface tends to zero rigorously. We prove convergence of the solutions of the Navier-Stokes/Allen-Cahn system to solutions of a sharp interface model, where the interface evolution is given by the mean curvature flow with an additional convection term coupled to a two-phase Navier-Stokes system with surface tension. This is done by constructing an approximate solution from the limiting system via matched asymptotic expansions together with a novel Ansatz for the highest order term, and then estimating its difference with the real solution with the aid of a refined spectral estimate of the linearized Allen-Cahn operator near the approximate solution.

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BibTeX RIS

Helmut Abels, Mingwen Fei. 2023-03-07. Sharp Interface Limit for a Navier-Stokes/Allen-Cahn System with Different Viscosities. https://arxiv.org/abs/2201.09343

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