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