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Existence of solutions for a higher order non-local equation appearing in crack dynamics

Mellet, Antoine; Imbert, Cyril (2011), Existence of solutions for a higher order non-local equation appearing in crack dynamics, Nonlinearity, 24, 12, p. Paper 3487. http://dx.doi.org/10.1088/0951-7715/24/12/008

Type
Article accepté pour publication ou publié
External document link
http://hal.archives-ouvertes.fr/hal-00451017/fr/
Date
2011
Journal name
Nonlinearity
Volume
24
Number
12
Publisher
IOP Science
Pages
Paper 3487
Publication identifier
Metadata
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Author(s)
Mellet, Antoine
Imbert, Cyril cc
Abstract (EN)
In this paper, we prove the existence of non-negative solutions for a non-local third order degenerate parabolic equation arising in the modeling of hydraulic fractures. The equation is similar to the well-known thin film equation, but the Laplace operator is replaced by a Dirichlet-to-Neumann type operator (which can be defined using the periodic Hilbert transform). The main difficulties are due to the fact that this equation is non-local, and that the natural energy estimates are not as good as in the case of the thin film equation.
Subjects / Keywords
Non-local equation; Higher order equation; Hydraulic fractures; periodic Hilbert transform; Non-negative solutions; Thin film equation