HAL : hal-00451017, version 1

arXiv : 1001.5105

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

Cyril Imbert 1, Antoine Mellet 2

(27/01/2010)

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.

1 :	CEntre de REcherches en MAthématiques de la DEcision (CEREMADE)
	CNRS : UMR7534 – Université Paris Dauphine - Paris IX
2 :	Department of Mathematics, University of Maryland
	University of Maryland

Domaine

:

Mathématiques/Equations aux dérivées partielles

Mots Clés : Hydraulic fractures – Higher order equation – Non-local equation – Thin film equation – Non-negative solutions – periodic Hilbert transform

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Contributeur : Cyril Imbert <>

Soumis le : Mercredi 27 Janvier 2010, 23:45:49

Dernière modification le : Jeudi 28 Janvier 2010, 06:29:55