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)

A proof of a new integral inequality for the Riesz transforms is presented, together with applications to obtain blow-up, in finite time, for a general class of initial data on a non-linear transport… (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)

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)

Here 2 > 0 is a (small) parameter, say 0 < 2 < 1, and Ω ⊂ <n is an open bounded domain. The nonnegative function F is a double well potential vanishing only for two values of u, say u = −1 and u =… (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)

During development, extracellular signaling molecules interact with intracellular gene networks to control the specification, pattern and size of organs. One such signaling molecule is Hedgehog (Hh).… (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)

This article emphasizes the role played by a remarkable pointwise inequality satisfied by fractionary derivatives in order to obtain maximum principles and Lp-decay of solutions of several… (More)

We study the initial value problem for dissipative 2D Quasi-geostrophic equations proving local existence, global results for small initial data in the super-critical case, decay of L-norms and… (More)