The Muskat problem models the filtration of two incompressible immiscible fluids of different characteristics in porous media. In this paper, we consider both the 2D and 3D setting of two fluids of different constant densities and different constant viscosities. In this situation, the related contour equations are non-local, not only in the evolution system, but also in the implicit relation between the amplitude of the vorticity and the free interface. Among other extra difficulties, no maximum principles are available for the amplitude and the slopes of the interface in L ∞ . We prove global in time existence results for medium size initial stable data in critical spaces. We also enhance previous methods by showing smoothing (instant analyticity), improving the medium size constant in 3D, together with sharp decay rates of analytic norms. The found technique is twofold, giving ill-posedness in unstable situations for very low regular solutions.

Abstract. We prove time decay of solutions to the Muskat equation in 2D and in 3D. In [11] and [12], the authors introduce the normsin order to prove global existence of solutions to the Muskat problem. In this paper, for the 3D Muskat problem, given initial data f 0 ∈ H l (R 2 ) for some l ≥ 3 such that f 0 1 < k 0 for a constant k 0 ≈ 1/5, we prove uniform in time bounds of f s(t) for −d < s < l − 1 and assuming f 0 ν < ∞ we prove time decay estimates of the form f s(t) (1 + t) −s+ν for 0 ≤ s ≤ l − 1 and −d ≤ ν < s. These large time decay rates are the same as the optimal rate for the linear Muskat equation. We also prove analogous results in 2D.

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In this paper, we consider sufficient conditions, called continuation criteria, for global existence and uniqueness of classical solutions to the three-dimensional relativistic Vlasov-Maxwell system. In the compact momentum support setting, we prove that p 18 5r −1+β 0 1 2 .