论文标题
关于不可压缩的Navier-Stokes和Allen-Cahn耦合系统的尖锐接口限制的评论
Remarks on Sharp Interface Limit for an Incompressible Navier-Stokes and Allen-Cahn Coupled System
论文作者
论文摘要
在本文中，我们关注了不可压缩的Navier-Stokes和Allen-Cahn耦合系统的急剧接口限制。当通过$ \ varepsilon $参数化的弥漫界面区域的厚度为零时，我们证明了不可压缩的Navier-Stokes和Allen-CaHn耦合系统的解决方案会收敛到$ l^\ infty（l^2）$ cap l^2（H^1）$ l^2（H^2（H^1）中， $ \ varepsilon $。证明由两个部分组成：一个是构造合适的近似解决方案，另一个是Sobolev空间中误差函数的估计。除了仔细的能量估计外，围绕近似解决方案的不可压缩的Navier-Stokes和Allen-CAHN耦合系统的线性化操作员的光谱估计基本用于得出误差函数的均匀估计。速度的收敛性很高，这是因为跨种族界面区域的速度层相对较弱。
We are concerned with the sharp interface limit for an incompressible Navier-Stokes and Allen-Cahn coupled system in this paper. When the thickness of the diffuse interfacial zone, which is parameterized by $\varepsilon$, goes to zero, we prove that a solution of the incompressible Navier-Stokes and Allen-Cahn coupled system converges to a solution of a sharp interface model in the $L^\infty(L^2)\cap L^2(H^1)$ sense on a uniform time interval independent of the small parameter $\varepsilon$. The proof consists of two parts: one is the construction of a suitable approximate solution and another is the estimate of the error functions in Sobolev spaces. Besides the careful energy estimates, a spectral estimate of the linearized operator for the incompressible Navier-Stokes and Allen-Cahn coupled system around the approximate solution is essentially used to derive the uniform estimates of the error functions. The convergence of the velocity is well expected due to the fact that the layer of the velocity across the diffuse interfacial zone is relatively weak.
