Self-similar solutions for a fractional thin film equation governing hydraulic fractures
نویسندگان
چکیده
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 ≥ 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 ∈ (0,∞) 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. AMS Classification: 35G30, 35R11, 35C06
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