# 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.
Keywords :
Type de document :
Article dans une revue
Communications in Mathematical Physics, Springer Verlag, 2015, 340 (3), pp.1187-1229 〈http://link.springer.com/article/10.1007/s00220-015-2459-9〉. 〈10.1007/s00220-015-2459-9〉
Liste complète des métadonnées

Littérature citée [21 références]

https://hal.archives-ouvertes.fr/hal-00967393
Contributeur : Cyril Imbert <>
Soumis le : samedi 2 janvier 2016 - 22:20:48
Dernière modification le : jeudi 11 janvier 2018 - 06:12:17

### Fichiers

self-crack-v3.pdf
Fichiers produits par l'(les) auteur(s)

### 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 〈http://link.springer.com/article/10.1007/s00220-015-2459-9〉. 〈10.1007/s00220-015-2459-9〉. 〈hal-00967393v2〉

### Métriques

Consultations de la notice

## 329

Téléchargements de fichiers