We study a 1D transport equation with nonlocal velocity and show the formation of singularities in finite time for a generic family of initial data. By adding a diffusion term the finite time… (More)

[1] The crustal structure of the northern Gulf of California transtensional margin has been investigated by a 280-km-long NW-SE profile, including deep multichannel seismic reflection and densely… (More)

We study the dynamics of the interface between two incompressible 2-D flows where the evolution equation is obtained from Darcy’s law. The free boundary is given by the discontinuity among the… (More)

The Muskat problem models the dynamics of the interface between two incompressible immiscible fluids with different constant densities. In this work we prove three results. First we prove an L(R)… (More)

We prove several weighted inequalities involving the Hilbert transform of a function f (x) and its derivative. One of those inequalities, − ∫ fx(x)[Hf (x) − Hf (0)] |x|α dx Cα ∫ (f (x) − f (0))2… (More)

We give conditions that rule out formation of sharp fronts for certain two-dimensional incompressible flows. We show that a necessary condition of having a sharp front is that the flow has to have… (More)

In this paper we study 1D equations with nonlocal flux. These models have resemblance of the 2D quasi-geostrophic equation. We show the existence of singularities in finite time and construct… (More)

We consider the problem of the evolution of the interface given by two incompressible fluids through a porous medium, which is known as the Muskat problem and in two dimensions it is mathematically… (More)

We study the free boundary evolution between two irrotational, incompressible and inviscid fluids in 2-D without surface tension. We prove local existence in Sobolev spaces when, initially, the… (More)