Details
| Original language | English |
|---|---|
| Pages (from-to) | 925-943 |
| Number of pages | 19 |
| Journal | Communications in Partial Differential Equations |
| Volume | 38 |
| Issue number | 5 |
| Publication status | Published - 10 Apr 2013 |
Abstract
We consider the Cahn-Hilliard equation on a manifold with conical singularities. We first show the existence of bounded imaginary powers for suitable closed extensions of the bilaplacian. Combining results and methods from singular analysis with a theorem of Clément and Li we then prove the short time solvability of the Cahn-Hilliard equation in Lp-Mellin-Sobolev spaces and obtain the asymptotics of the solution near the conical points. We deduce, in particular, that regularity is preserved on the smooth part of the manifold and singularities remain confined to the conical points. We finally show how the Allen-Cahn equation can be treated by simpler considerations. Again we obtain short time solvability and the behavior near the conical points.
Keywords
- Allen-Cahn equation, Cahn-Hilliard equation, Cone pseudodifferential operators, Conical singularities, Mellin-Sobolev spaces, Short time asymptotics
ASJC Scopus subject areas
- Mathematics(all)
- Analysis
- Mathematics(all)
- Applied Mathematics
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In: Communications in Partial Differential Equations, Vol. 38, No. 5, 10.04.2013, p. 925-943.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - The Cahn-Hilliard Equation and the Allen-Cahn Equation on Manifolds with Conical Singularities
AU - Roidos, Nikolaos
AU - Schrohe, Elmar
N1 - Copyright: Copyright 2013 Elsevier B.V., All rights reserved.
PY - 2013/4/10
Y1 - 2013/4/10
N2 - We consider the Cahn-Hilliard equation on a manifold with conical singularities. We first show the existence of bounded imaginary powers for suitable closed extensions of the bilaplacian. Combining results and methods from singular analysis with a theorem of Clément and Li we then prove the short time solvability of the Cahn-Hilliard equation in Lp-Mellin-Sobolev spaces and obtain the asymptotics of the solution near the conical points. We deduce, in particular, that regularity is preserved on the smooth part of the manifold and singularities remain confined to the conical points. We finally show how the Allen-Cahn equation can be treated by simpler considerations. Again we obtain short time solvability and the behavior near the conical points.
AB - We consider the Cahn-Hilliard equation on a manifold with conical singularities. We first show the existence of bounded imaginary powers for suitable closed extensions of the bilaplacian. Combining results and methods from singular analysis with a theorem of Clément and Li we then prove the short time solvability of the Cahn-Hilliard equation in Lp-Mellin-Sobolev spaces and obtain the asymptotics of the solution near the conical points. We deduce, in particular, that regularity is preserved on the smooth part of the manifold and singularities remain confined to the conical points. We finally show how the Allen-Cahn equation can be treated by simpler considerations. Again we obtain short time solvability and the behavior near the conical points.
KW - Allen-Cahn equation
KW - Cahn-Hilliard equation
KW - Cone pseudodifferential operators
KW - Conical singularities
KW - Mellin-Sobolev spaces
KW - Short time asymptotics
UR - http://www.scopus.com/inward/record.url?scp=84876103673&partnerID=8YFLogxK
U2 - 10.1080/03605302.2012.736913
DO - 10.1080/03605302.2012.736913
M3 - Article
AN - SCOPUS:84876103673
VL - 38
SP - 925
EP - 943
JO - Communications in Partial Differential Equations
JF - Communications in Partial Differential Equations
SN - 0360-5302
IS - 5
ER -