We paralinearize the Muskat equation to extract an explicit parabolic evolution equation having a compact form. This result is applied to give a simple proof of the local well-posedness of the Cauchy problem for rough initial data, in homogeneous Sobolev spacesḢ 1 (R) ∩Ḣ s (R) with s > 3/2. This paper is essentially self-contained and does not rely on general results from paradifferential calculus.

We prove a global existence result of a unique strong solution in Ḣ5/2 ∩ Ḣ3/2 with small Ḣ3/2 norm for the 2D stable Muskat problem, hence allowing the interface to have arbitrary large finite slopes and finite energy (thanks to the L 2 maximum principle). The proof is based on the use of a new formulation of the Muskat equation that involves oscillatory terms. Then, a careful use of interpolation inequalities in homogeneneous Besov spaces allows us to close the a priori estimates.

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We review some recent results on the Muskat problem modelling multiphase flow in porous media. Furthermore, we prove a new regularity criteria in terms of some norms of the initial data in critical spaces (Ẇ 1,∞ andḢ 3/2 ).2010 Mathematics Subject Classification. Primary 35A01, 35D30, 35D35, 35Q35, 35Q86.

In this article, we study the critical dissipative surface quasi-geostrophic equation (SQG) in R 2 . Motivated by the study of the homogeneous statistical solutions of this equation, we show that for any large initial data θ 0 liying in the space Λ s (H s uloc (R 2 )) ∩ L ∞ (R 2 ) the critical (SQG) has a global weak solution in time for 1/2 < s < 1. Our proof is based on an energy inequality verified the (SQG) R, equation which is nothing but the equation (SQG) equation with truncated and regularized initial data. By classical compactness arguments, we show that we are able to pass to the limit (R → ∞, → 0) in (SQG) R, and that the limit solution has the desired regularity.