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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 asymptotic behavior of viscosity solution in the critical case. Our proofs are based on a maximum principle valid for more general flows.
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.
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)
A technique is introduced to relate differentiation and covering properties of a basis. In particular, we find that the basis associated with a sparse set of directions differentiates integrals of functions locally in L(2).
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)
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)
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)