标题： 粘性不可压缩流体经典两相流动的Navier-Stokes/Allen-Cahn系统逼近 显示英文标题

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

摘要： 我们证明了Navier-Stokes/Allen-Cahn系统在二维和三维空间中，当光滑有界域中存在极限系统的光滑解时，收敛到两种粘性不可压缩流体具有相同粘度的两相流动的经典尖界面模型。 此外，我们通过相对熵方法得到了误差估计。 我们的结果成立当Allen-Cahn方程中的迁移率$m_\varepsilon>0$以亚临界方式趋于零，即$m_\varepsilon= m_0 \varepsilon^\beta$对于某个$\beta\in (0,2)$和$m_0>0$。 证明是通过相对熵论证来显示Navier-Stokes/Allen-Cahn系统的解保持与修正后的两相流动问题的解接近，该问题在界面运动中增加了额外的平均曲率流项$m_\varepsilon H_{\Gamma_t}$。 在第二步中，很容易看出修正后问题的解与原始两相流动的解接近。 显示英文摘要

摘要： 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 $m_\varepsilon>0$ in the Allen-Cahn equation tends to zero in a subcritical way, i.e., $m_\varepsilon= m_0 \varepsilon^\beta$ for some $\beta\in (0,2)$ and $m_0>0$. 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 $m_\varepsilon H_{\Gamma_t}$ 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.