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Well-posedness of a fractional porous medium equation on an evolving surface
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Alphonse, Amal and Elliott, Charles M. (2016) Well-posedness of a fractional porous medium equation on an evolving surface. Nonlinear Analysis: Theory, Methods & Applications, 137 . pp. 3-42. doi:10.1016/j.na.2016.01.010 ISSN 0362-546X.
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Official URL: http://dx.doi.org/10.1016/j.na.2016.01.010
Abstract
We investigate the existence, uniqueness, and L1-contractivity of weak solutions to a porous medium equation with fractional diffusion on an evolving hypersurface. To settle the existence, we reformulate the equation as a local problem on a semi-infinite cylinder, regularise the porous medium nonlinearity and truncate the cylinder. Then we pass to the limit first in the truncation parameter and then in the nonlinearity, and the identification of limits is done using the theory of subdifferentials of convex functionals. In order to facilitate all of this, we begin by studying (in the setting of closed Riemannian manifolds and Sobolev spaces) the fractional Laplace–Beltrami operator which can be seen as the Dirichlet-to-Neumann map of a harmonic extension problem. A truncated harmonic extension problem will also be examined and convergence results to the solution of the harmonic extension will be given. For a technical reason, we will also consider some related extension problems on evolving hypersurfaces which will provide us with the minimal time regularity required on the harmonic extensions in order to properly formulate the moving domain problem. This functional analytic theory is of course independent of the fractional porous medium equation and will be of use generally in the analysis of fractional elliptic and parabolic problems on manifolds.
Item Type: | Journal Article | ||||||||||
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Subjects: | Q Science > QA Mathematics | ||||||||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||||||
Library of Congress Subject Headings (LCSH): | Evolution equations, Nonlinear , Riemannian manifolds, Sobolev spaces | ||||||||||
Journal or Publication Title: | Nonlinear Analysis: Theory, Methods & Applications | ||||||||||
Publisher: | Pergamon-Elsevier Science Ltd | ||||||||||
ISSN: | 0362-546X | ||||||||||
Official Date: | May 2016 | ||||||||||
Dates: |
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Volume: | 137 | ||||||||||
Page Range: | pp. 3-42 | ||||||||||
DOI: | 10.1016/j.na.2016.01.010 | ||||||||||
Status: | Peer Reviewed | ||||||||||
Publication Status: | Published | ||||||||||
Access rights to Published version: | Restricted or Subscription Access | ||||||||||
Date of first compliant deposit: | 21 June 2016 | ||||||||||
Date of first compliant Open Access: | 12 February 2017 | ||||||||||
Funder: | Engineering and Physical Sciences Research Council (EPSRC) | ||||||||||
Grant number: | EP/H023364/1 | ||||||||||
Adapted As: |
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