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April  2010, 26(2): 431-453. doi: 10.3934/dcds.2010.26.431

The surface diffusion flow on rough phase spaces

Joachim Escher 1, and  Piotr B. Mucha 2,

1. 

Institute for Applied Mathematics, Leibniz University of Hanover, D-30167 Hanover
2. 

Institute of Applied Mathematics and Mechanics, University of Warsaw, 02-097 Warszawa, Poland

Received  July 2008 Revised  July 2009 Published  October 2009

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.
Keywords: maximal regularity, centre manifold, stability, Besov spaces, Surface diffusion flow, free boundary problem., global existence, geometric evolution equation.
Mathematics Subject Classification: 35R35, 35K55, 35S30, 80A2.
Citation: Joachim Escher, Piotr B. Mucha. The surface diffusion flow on rough phase spaces. Discrete & Continuous Dynamical Systems, 2010, 26 (2) : 431-453. doi: 10.3934/dcds.2010.26.431
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