# Self-similar solutions for a fractional thin film equation governing hydraulic fractures

Abstract : In this paper, self-similar solutions for a fractional thin film equation governing hydraulic fractures are constructed. One of the boundary conditions, which accounts for the energy required to break the rock, involves the toughness coefficient $K\geq 0$. Mathematically, this condition plays the same role as the contact angle condition in the thin film equation. We consider two situations: The zero toughness ($K=0$) and the finite toughness $K\in(0,\infty)$ cases. In the first case, we prove the existence of self-similar solutions with constant mass. In the second case, we prove that for all $K>0$ there exists an injection rate for the fluid such that self-similar solutions exist.
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https://hal.archives-ouvertes.fr/hal-00967393
Contributor : Cyril Imbert <>
Submitted on : Saturday, January 2, 2016 - 10:20:48 PM
Last modification on : Thursday, March 19, 2020 - 12:26:02 PM

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### Citation

Cyril Imbert, Antoine Mellet. Self-similar solutions for a fractional thin film equation governing hydraulic fractures. Communications in Mathematical Physics, Springer Verlag, 2015, 340 (3), pp.1187-1229 ⟨10.1007/s00220-015-2459-9⟩. ⟨hal-00967393v2⟩

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