2016/11/14 by Helmut Abels, Yuning Liu, Abels, Helmut +1 · 1 citation
Computer Science · Materials Science · Physics and Astronomy · #35Q30 #35Q35 #35R35 #76D05 #76D45 #76T99 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Fluid Dynamics (physics.flu-dyn) #Solidification and crystal growth phenomena #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.1611.04422
openalex publication_date 2016/11/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the sharp interface limit of a coupled Stokes/Allen-Cahn system, when a parameter ε>0 that is proportional to the thickness of the diffuse interface tends to zero, in a two dimensional bounded domain. For sufficiently small times we prove convergence of the solutions of the Stokes/Allen-Cahn system to solutions of a sharp interface model, where the interface evolution is given by the mean curvature equation with an additional convection term coupled to a two-phase Stokes system with an additional contribution to the stress tensor, which describes the capillary stress. To this end we construct a suitable approximation of the solution of the Stokes/Allen-Cahn system, using three levels of the terms in the formally matched asymptotic calculations, and estimate the difference with the aid of a suitable refinement of a spectral estimate of the linearized Allen-Cahn operator. Moreover, a careful treatment of the coupling terms is needed.