强FCH流的正则化曲线长度
Regularized Curve Lengthening from the Strong FCH Flow
摘要：
我们提出了一个严格的分析下薄的接口限制，$\varepsilon\ll1$，质量保持$L^2$ -梯度流的功能化的Cahn-Hilliard方程的强标度的近圆形双层界面的瞬态演化。对于一个域$Ω\subset{\mathbb R}^2$，我们构建了一个双层流形与边界组成的准平衡流和投影到流形上，关联功能$u$在一个$H^2$管状附近的流形与接口$Γ$嵌入在$Ω$。这些接口，因此双层流形，参数化的有限，但渐近大量的自由度。流形包含一个独特的，平移和质量约束，平衡的梯度流，其投影界面是圆形的指数小的修正。细管状邻域是向前不变的流动下的轨道，最终收敛到平衡。这些轨道的投影产生一个界面的演变相当于在领导秩序的正常运动{ \sl对}平均曲率，正则化的高阶Willmore表达式的正则化曲线延长运动。曲线拉长是由从远离界面的$Ω$区域吸收过量质量驱动的，通常导致在$\varepsilon\to0$极限中不适定的非平凡动力学。
英文摘要：
We present a rigorous analysis of the transient evolution of nearly circular bilayer interfaces evolving under the thin interface limit, $\varepsilon\ll1$, of the mass preserving $L^2$-gradient flow of the strong scaling of the functionalized Cahn-Hilliard equation. For a domain $Ω\subset{\mathbb R}^2$ we construct a bilayer manifold with boundary comprised of quasi-equilibrium of the flow and a projection onto the manifold that associates functions $u$ in an $H^2$ tubular neighborhood of the manifold with an interface $Γ$ embedded in $Ω$. These interfaces, and hence the bilayer manifold, are parameterized by a finite but asymptotically large number of degrees of freedom. The manifold contains a unique, up to translation and mass constraint, equilibrium of the gradient flow whose projected interface is circular up to exponentially small corrections. The thin tubular neighborhood is forward invariant under the flow with orbits that ultimately converge to the equilibrium. Projections of these orbits yield an interfacial evolution equivalent at leading order to the regularized curve-lengthening motion characterized by normal motion {\sl against} mean curvature, regularized by a higher order Willmore expression. The curve lengthening is driven by absorption of excess mass from the regions of $Ω$ away from the interface, generically leading to nontrivial dynamics that are ill-posed in the $\varepsilon\to0$ limit.
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