TY - JOUR

T1 - The surface diffusion flow on rough phase spaces

AU - Escher, Joachim

AU - Mucha, Piotr B.

PY - 2010/4

Y1 - 2010/4

N2 - 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.

AB - 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.

KW - Besov spaces

KW - Centre manifold

KW - Free boundary problem

KW - Geometric evolution equation

KW - Global existence

KW - Maximal regularity

KW - Stability

KW - Surface diffusion flow

UR - http://www.scopus.com/inward/record.url?scp=77949943046&partnerID=8YFLogxK

U2 - 10.3934/dcds.2010.26.431

DO - 10.3934/dcds.2010.26.431

M3 - Article

AN - SCOPUS:77949943046

VL - 26

SP - 431

EP - 453

JO - Discrete and Continuous Dynamical Systems

JF - Discrete and Continuous Dynamical Systems

SN - 1078-0947

IS - 2

ER -