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Self-Similar Solutions for a Fractional Thin Film Equation Governing Hydraulic Fractures

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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 ≥ 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.

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Authors and Affiliations

  1. CNRS, UMR 7580, Université Paris-Est Créteil, 61 avenue du Général de Gaulle, 94 010, Créteil Cedex, France

    C. Imbert

  2. Department of Mathematics, University of Maryland, College Park, MD, 20742, USA

    A. Mellet

  3. Fondation Sciences Mathématiques de Paris, 11 rue Pierre et Marie Curie, 75231, Paris Cedex 05, France

    A. Mellet

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  1. C. Imbert
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  2. A. Mellet
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Correspondence to A. Mellet.

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Communicated by L. Caffarelli

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Imbert, C., Mellet, A. Self-Similar Solutions for a Fractional Thin Film Equation Governing Hydraulic Fractures. Commun. Math. Phys. 340, 1187–1229 (2015). https://doi.org/10.1007/s00220-015-2459-9

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  • DOI: https://doi.org/10.1007/s00220-015-2459-9

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