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u(x, 0) = u0(x), where u0 ∈ H(R). Our principal aim here is to lower the best index s for which one has local well posedness in H(R), i.e. existence, uniqueness, persistence and continuous dependence on the data, for a finite time interval, whose size depends on ‖u0‖Hs . Equation in (1.1) was derived by Korteweg and de Vries [21] as a model for long wave(More)
The KdV equation, which was first derived as a model for unidirectional propagation of nonlinear dispersive long waves [21], has been considered in different contexts, namely in its relation with the inverse scattering method, in plasma physics, and in algebraic geometry (see [24], and references therein). Our purpose is to study local and global(More)
The purpose of this paper is to investigate bilinear variants of the restriction and Kakeya conjectures, to relate them to the standard formulations of these conjectures, and to give applications of this bilinear approach to existing conjectures. The methods used are based on several observations and results of Bourgain (see [2]-[6]), together with some(More)
In this paper we address the question of the singular vortex dynamics exhibited in [15], which generates a corner in finite time. The purpose is to prove that under some appropriate small regular perturbation the corner still remains. Our approach uses the Hasimoto transform and deals with the long range scattering properties of a GrossPitaevski equation(More)
I’ll present some recent work with F. Planchon on bilinear virial identities for the nonlinear Schrodinger equation, which are extensions of the Morawetz interaction inequalities firstly obtained by Colliander, Keel, Staffilani, Takaoka and Tao. We recover and extend known bilinear improvements to Strichartz inequalities and provide applications to various(More)
Spectral properties and the confinement phenomenon for the couplingH+V are studied, where H = −iα ·∇+mβ is the free Dirac operator in R and V is a measure-valued potential. The potentials V under consideration are given in terms of surface measures on the boundary of bounded regular domains in R. A criterion for the existence of point spectrum is given,(More)
Sound localization is the ability to pinpoint the direction a sound is coming from based on auditory cues alone. Neurons in the brain which mediate this behavior are active only when sound comes from a particular direction. This thesis uses physiological and anatomical methods to investigate the computations which lead to such space-specific neural(More)