Uniqueness of a finite-toughness hydraulic-fracture similarity profile
Fix $K > 0$. For $x, z \in (-1, 1)$ let $$G(x,z) = \frac{1}{\pi} \log \frac{1 - xz + \sqrt{(1 - x^2)(1 - z^2)}}{|x - z|}, \qquad g(x,z) = \tfrac12 \bigl[(z - x) G(x,z) + (z + x) G(x,-z)\bigr],$$ the products at $z = \pm x$ being given their continuous limits, and $g(\pm 1, z) = 0$. For an even continuous function $U \colon \mathbb{R} \to [0, \infty)$ supported in $[-1, 1]$ and positive on $(-1, 1)$, let $\mathcal{T}_U(z) = \int_z^1 U(y)\,dy$ for $z > 0$ and $\mathcal{T}_U(z) = -\int_{-1}^z U(y)\,dy$ for $z < 0$.
Here $G$ is the Green function of $(-\Delta)^{1/2}$ on $(-1, 1)$ and $g$ that of $U \mapsto I(U)'$, where $I = -(-\Delta)^{1/2}$. The equation below is the integral form, found by Imbert and Mellet, of $U^3 I(U)' = xU + \tfrac32 \mathcal{T}_U$ on $(-1, 1)$, the equation for the normalised profile $U$ of a self-similar solution $u(t, x) = t^{1/3} U(t^{-2/3} x)$ of the fractional thin-film equation $\partial_t u + \partial_x \bigl(u^3 \partial_x I(u)\bigr) = h(t)\delta$ for a hydraulic fracture driven by fluid injected at the origin, whose square-root toughness condition at the tips becomes the first condition below. Imbert and Mellet proved that for every $K > 0$ there is such a profile, which needs a constant injection rate $h$; the fracture length then grows like $t^{2/3}$ and its opening like $t^{1/3}$.
Problem. Is there at most one such $U$ satisfying $$\lim_{x \to \pm 1,\ |x| < 1} \frac{U(x)}{\sqrt{1 - x^2}} = K \qquad\text{and}\qquad U(x) = K\sqrt{1 - x^2} + \int_{-1}^1 g(x,z)\, \frac{zU(z) + \tfrac32 \mathcal{T}_U(z)}{U(z)^3}\, dz \quad (-1 < x < 1),$$ with all integrals absolutely convergent Lebesgue integrals?
Source: Imbert and Mellet (2015), Remark 1.2(4), who ask for uniqueness of the self-similar solutions for zero and for positive toughness; stated for the physical mobility exponent 3, positive toughness and the normalised integral equation by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, fractional thin-film equations, hydraulic fractures, self-similar solutions
- Literature checked
- Nov. 5, 2026 by clemens-admin
References
@article{imbert2015,
author = {Imbert, Cyril and Mellet, Antoine},
title = {Self-similar solutions for a fractional thin film equation governing hydraulic fractures},
journal = {Comm. Math. Phys.},
volume = {340},
number = {3},
year = {2015},
pages = {1187--1229},
doi = {10.1007/s00220-015-2459-9},
eprint = {1403.7491},
archivePrefix = {arXiv},
}
@misc{aim-uniqueness-of-a-finite-toughness-hydraulic-fracture-similarity-profile,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Uniqueness of a finite-toughness hydraulic-fracture similarity profile}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/4f5b1659355cb61d194da94ef14615214e1363a8/problems/755-uniqueness-of-a-finite-toughness-hydraulic-fracture-similarity-profile.md},
note = {Entry 755 at commit 4f5b165, 2026-11-04},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-01166, "Uniqueness of a finite-toughness hydraulic-fracture similarity profile", clopen.solutions.