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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)
In a number of useful applications, e.g., data compression, the appropriate partial sums of the Fourier series are formed by taking into consideration the size of the coefficients rather than the size of the frequencies involved. The purpose of this paper is to show the limitations of that method of summation. We use several results from the number theory… (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)
I discuss the impact various papers have had on my own work.
The appearance of fluid filaments during the evolution of a viscous fluid jet is a commonly observed phenomenon. It is shown here that the break-up of such a jet subject to capillary forces is impossible through the collapse of a uniform filament.
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)