# Fractional equation of thin films for hydraulic fractures

Abstract : In this thesis, we study two degenerate, non-local parabolic equations, a fractional thin film equation and a fractional porous medium equation. The introduction contains a presentation of problems, the previous results in the literature and a brief presentation of our results. In the second chapter, we present a short overview of the De Giorgi method used to prove Hölder regularity of solutions of elliptic equations. Moreover, we present the results using this approach in the local and non-local parabolic cases. In the third chapter we prove existence of weak solutions of a fractional thin film equation. It is a non-local degenerate parabolic equation of order $alpha + 2$ where $0 < alpha < 2$. It is a generalization of an equation studied by Imbert and Mellet in 2011 for $alpha = 1$. To construct these solutions, we consider a regularized problem then we pass to the limit using Sobolev embedding theorem, that's why we distinguish two cases $0 < alpha < 1$ and $1 leq alpha < 2$. We also prove that the solution is positive if the initial condition is so. The fourth chapter is dedicated for a fractional porous medium equation. We prove Hölder regularity of positive weak solutions satisfying energy estimates. First, we prove the existence of weak solutions that satisfy energy estimates. We distiguish two cases $0 < alpha < 1$ and $1 leq alpha < 2$ because of divergence problems. The we prove De Giorgi Lemmas about oscillation reduction from above and from below. This is not suffisant. We need to improve the lemma about oscillation reduction from above. So we pass by an intermediate values lemma and we prove an improved oscillation reduction lemma from above. Finally, we prove Hölder regularity of solutions using the scaling property
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https://tel.archives-ouvertes.fr/tel-02066162
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Submitted on : Wednesday, March 13, 2019 - 11:11:07 AM
Last modification on : Thursday, March 14, 2019 - 1:08:13 AM

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TH2018PESC1061.pdf
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• HAL Id : tel-02066162, version 1

### Citation

Rana Tarhini. Fractional equation of thin films for hydraulic fractures. General Mathematics [math.GM]. Université Paris-Est, 2018. English. ⟨NNT : 2018PESC1061⟩. ⟨tel-02066162⟩

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