论文标题
系统耦合缩放平均曲率流和扩散的定性特性
Qualitative Properties for a System Coupling Scaled Mean Curvature Flow and Diffusion
论文作者
论文摘要
我们考虑了一个由超表面的几何进化方程组成的系统，以及在这种不断发展的超表面上的抛物线方程。更确切地说，我们讨论了用术语缩放的平均曲率流，该术语取决于在表面上定义的数量，该量与该数量的扩散方程相连。分析了解决方案的几种属性。重点放在我们定性中的表面在何种程度上演变出与通常的平均曲率流相似的演变。为此，我们表明表面积严格降低，但给出了一个无限时间的表面的例子。此外，平均凸度是保守的，而凸度不是。最后，我们构建了一个嵌入式的超脸，随着时间的流逝，它会发展出一种自我交流。此外，还包括如何将我们的方程式解释为梯度流的正式解释。
We consider a system consisting of a geometric evolution equation for a hypersurface and a parabolic equation on this evolving hypersurface. More precisely, we discuss mean curvature flow scaled with a term that depends on a quantity defined on the surface coupled to a diffusion equation for that quantity. Several properties of solutions are analyzed. Emphasis is placed on to what extent the surface in our setting qualitatively evolves similar as for the usual mean curvature flow. To this end, we show that the surface area is strictly decreasing but give an example of a surface that exists for infinite times nevertheless. Moreover, mean convexity is conserved whereas convexity is not. Finally, we construct an embedded hypersurface that develops a self-intersection in the course of time. Additionally, a formal explanation of how our equations can be interpreted as a gradient flow is included.
