The surface diffusion flow on rough phase spaces
The surface diffusion flow on rough phase spaces
复制标题
粗糙相空间上的表面扩散流
DOI：
10.3934/dcds.2010.26.431
复制
发表时间：
2009
影响因子：
1.1
通讯作者：
P. Mucha
中科院分区：
文献类型：
--
作者：
J. Escher;P. Mucha
表面扩散流是紧超曲面对H^{-1}$内积的表面泛函的梯度流，并导致一个四阶非线性演化方程。这是一个本质上困难的问题，由于缺乏极大值原理，并且已知该流可以在有限时间内将平滑嵌入的均匀凸初始曲面驱动为非凸曲面，然后再形成奇点[15,16]。另一方面，我们也知道，对于非凸初始数据的解，奇点可能在有限时间内出现，参见[10]。
The surface diffusion flow is the gradient flow of the surface functional of compact hypersurfaces with respect to the inner product of $H^{-1}$ and leads to a nonlinear evolution equation of fourth order. This is an intrinsically difficult problem, due to the lack of an maximum principle and it is known that this flow may drive smoothly embedded uniformly convex initial surfaces in finite time into non-convex surfaces before developing a singularity [15, 16]. On the other hand it also known that singularities may occur in finite time for solutions emerging from non-convex initial data, cf. [10].
Combining tools from harmonic analysis, such as Besov spaces, multiplier results with abstract results from the theory of maximal regularity we present an analytic framework in which we can investigate weak solutions to the original evolution equation. This approach allows us to prove well-posedness on a large (Besov) space of initial data which is in general larger than $C^2$ (and which is in the distributional sense almost optimal). Our second main result shows that the set of all compact embedded equilibria, i.e. the set of all spheres, is an invariant manifold in this phase space which attracts all solutions which are close enough (which respect to the norm of the phase space) to this manifold. As a consequence we
are able to construct non-convex initial data which generate global solutions, converging finally to a sphere.
关于中间表面扩散流的极限运动和自相交，
DOI：
--
发表时间：
2002
期刊：
J. of Evolution Equations 2
影响因子：
--
作者：
J.Escher;Y.Giga;K.Ito
通讯作者：
K.Ito