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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 interesting partial differential equations. In particular, there are applications to quasigeostrophic flows, in two space variables with critical viscosity, that model(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 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)
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 |x|1+α dx, is used to show finite time blow-up for a transport equation with nonlocal velocity. © 2006 Elsevier Masson SAS. All rights reserved. Résumé Dans cet(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)
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 = +1, and the minimizers u under consideration will take values precissely at that interval (−1 ≤ u ≤ +1). A description of the physical model can be found in(More)