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 singularity is prevented and the solutions exist globally in time.
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 explicit solutions to the equations where the singularities formed are shocks. For the critical viscosity case we show formation of singularities and global existence… (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). Hh is known to act as a morphogen, instructing different fates depending on the distance to its source. However, how Hh, when signaling across a cell field,… (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 densities and viscosities of the fluids. This physical scenario is known as the two dimensional Muskat problem or the two-phase Hele-Shaw flow. We prove… (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 difference of the gradients of the pressure in the normal direction has the proper sign, an assumption which is also known as the Rayleigh-Taylor condition. The… (More)
We analyze a distributed information network in which each node has access to the information contained in a limited set of nodes (its neighborhood) at a given time. A collective computation is carried out in which each node calculates a value that implies all information contained in the network (in our case, the average value of a variable that can take… (More)
Extended Abstract Gene regulatory networks set a second order approximation to genetics understanding, where the first order is the knowledge at the single gene activity level. With the increasing number of sequenced genomes, including human's, the time has come to investigate the interactions among myriads of genes that result into complex behaviours. The… (More)