Details
| Original language | English |
|---|---|
| Pages (from-to) | 431-453 |
| Number of pages | 23 |
| Journal | Discrete and Continuous Dynamical Systems |
| Volume | 26 |
| Issue number | 2 |
| Publication status | Published - Apr 2010 |
Abstract
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
- Besov spaces, Centre manifold, Free boundary problem, Geometric evolution equation, Global existence, Maximal regularity, Stability, Surface diffusion flow
ASJC Scopus subject areas
- Mathematics(all)
- Analysis
- Mathematics(all)
- Discrete Mathematics and Combinatorics
- Mathematics(all)
- Applied Mathematics
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In: Discrete and Continuous Dynamical Systems, Vol. 26, No. 2, 04.2010, p. 431-453.
Research output: Contribution to journal › Article › Research › peer review
}
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 -