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
TypeArticle accepté pour publication ou publié
External document linkhttp://hal.archives-ouvertes.fr/hal-00451017/fr/
MetadataShow full item record
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 / KeywordsNon-local equation; Higher order equation; Hydraulic fractures; periodic Hilbert transform; Non-negative solutions; Thin film equation
Showing items related by title and author.
Monteillet, Aurélien; Ley, Olivier; Cardaliaguet, Pierre; Barles, Guy (2009) Article accepté pour publication ou publié
Homogenization of First-Order Equations with (u/ε) -Periodic Hamiltonians. Part II: application to dislocations dynamics Imbert, Cyril; Monneau, Régis; Rouy, Elisabeth (2007) Document de travail / Working paper