HAL : hal-00451017, version 2

arXiv : 1001.5105

DOI : 10.1088/0951-7715/24/12/008

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Nonlinearity 24 (2001) 3487-3514

Versions disponibles :

v1 (28-01-2010)

v2 (06-05-2010)

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

Cyril Imbert 1, Antoine Mellet 2

(04/11/2001)

In this paper, we prove the existence of non-negative solutions for a non-local higher 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 operator, corresponding to the square root of the Laplace operator on a bounded domain with Neumann boundary conditions (which can also be defined using the periodic Hilbert transform). In our study, we have to deal with the usual difficulty associated to higher order equations (e.g. lack of maximum principle). However, there are important differences with, for instance, the thin film equation: First, our equation is nonlocal; Also the natural energy estimate is not as good as in the case of the thin film equation, and does not yields, for instance, boundedness and continuity of the solutions (our case is critical in dimension $1$ in that respect).

1 :	CEntre de REcherches en MAthématiques de la DEcision (CEREMADE)
	CNRS : UMR7534 – Université Paris IX - Paris Dauphine
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

hal-00451017, version 2

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Contributeur : Cyril Imbert <>

Soumis le : Vendredi 30 Avril 2010, 16:57:08

Dernière modification le : Lundi 14 Novembre 2011, 10:50:35