We study the dynamics of the interface between two incompressible fluids in a twodimensional porous medium whose flow is modeled by the Muskat equations. For the two-phase Muskat problem, we establish global well-posedness and decay to equilibrium for small H 2 perturbations of the rest state. For the one-phase Muskat problem, we prove local well-posedness for H 2 initial data of arbitrary size. Finally, we show that solutions to the Muskat equations instantaneously become infinitely smooth.≤ C(|h 1 | 2 + |h 2 | 2 )|h| 1.75 + C(|h 1 | 1.75 + |h 2 | 1.75 + 1)|h| 2 , so we conclude that ∇Q 1.25,± + w 1.25,± ≤ C

This work studies a simplified model of the gravitational instability of an initially homogeneous infinite medium, represented by $\TT^d$, based on the approximation that the mean fluid velocity is always proportional to the local acceleration. It is shown that, mathematically, this assumption leads to the restricted Patlak-Keller-Segel model considered by J\"ager and Luckhaus or, equivalently, the Smoluchowski equation describing the motion of self-gravitating Brownian particles, coupled to the modified Newtonian potential that is appropriate for an infinite mass distribution. We discuss some of the fundamental properties of a non-local generalization of this model where the effective pressure force is given by a fractional Laplacian with $0<\alpha<2$, and illustrate them by means of numerical simulations. Local well-posedness in Sobolev spaces is proven, and we show the smoothing effect of our equation, as well as a \emph{Beale-Kato-Majda}-type criterion in terms of $\rhomax$. It is also shown that the problem is ill-posed in Sobolev spaces when it is considered backward in time. Finally, we prove that, in the critical case (one conservative and one dissipative derivative), $\rhomax(t)$ is uniformly bounded in terms of the initial data for sufficiently large pressure forces.Comment: Accepted in Physica D: Nonlinear Phenomen