两亲双层膜的流形：边界稳定性
Manifolds of Amphiphilic Bilayers: Stability up to the Boundary
摘要：
我们认为质量保持$L^2$ -梯度流的功能化Cahn Hilliard梯度流的强标度和建立的准平衡双层\muckmucks的流形的边界上的非线性稳定性。在薄的但非零的界面宽度的限制，$\varepsilon\ll1,$的双层流形参数化的曲折模式，描述的界面演变和“珍珠”模式，控制界面附近的轮廓的结构。珠化模式是弱阻尼的，并且可以导致界面的动态破裂。两亲性界面可以延长以降低能量。我们引入一个隐式定义的参数化的界面形状，解耦这种增长的参数描述的形状，并引入一个非线性投影到流形从周围的邻居。双层流形具有渐近大但有限的尺寸，调整到最大限度地提高正常的曲率，同时保持曲折和珍珠模式之间的波数差距。模珍珠稳定性假设，我们表明，流形吸引附近的轨道到一个管状邻域本身，只要界面形状保持足够光滑，远离自相交。在配套文件中，arXiv：1907.02196，我们确定了开放的初始数据集，其轨道收敛到圆形平衡后，一个显着的瞬态，并推导出一个奇异摄动的界面演化组成的运动对曲率正则化的渐近弱Willmore项。
英文摘要：
We consider the mass preserving $L^2$-gradient flow of the strong scaling of the functionalized Cahn Hilliard gradient flow and establish the nonlinear stability of a manifold comprised of quasi-equilibrium bilayer \muckmucks up to the manifold's boundary. In the limit of thin but non-zero interfacial width, $\varepsilon\ll1,$ the bilayer manifold is parameterized by meandering modes that describe the interfacial evolution and "pearling" modes that control the structure of the profile near the interface. The pearling modes are weakly damped and can lead to the dynamic rupture of the interface. Amphiphilic interfaces can lengthen to decrease energy. We introduce an implicitly defined parameterization of the interfacial shape that uncouples this growth from the parameters describing the shape and introduce a nonlinear projection onto the manifold from a surrounding neighborhood. The bilayer manifold has asymptotically large but finite dimension tuned to maximize normal coercivity while preserving the wave-number gap between the meandering and the pearling modes. Modulo a pearling stability assumption, we show that the manifold attracts nearby orbits into a tubular neighborhood about itself so long as the interfacial shape remains sufficiently smooth and far from self-intersection. In a companion paper, arXiv:1907.02196, we identify open sets of initial data whose orbits converge to circular equilibrium after a significant transient, and derive a singularly perturbed interfacial evolution comprised of motion against curvature regularized by an asymptotically weak Willmore term.
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