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

Cyril Imbert (1) , Antoine Mellet (2)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 Department of Mathematics [College Park]

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.

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

Domaines: Equations aux dérivées partielles [math.AP]

Cyril Imbert, Antoine Mellet. Existence of solutions for a third order non-local equation appearing in crack dynamics. 2010. ⟨hal-00451017v1⟩