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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)

- TERENCE TAO, LUIS VEGA, Tony Carbery, Adela Moyua
- 1998

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)

We prove some sharp Hardy type inequalities related to the Dirac operator by elementary, direct methods. Some of these inequalities have been obtained previously using spectral information about the Dirac-Coulomb operator. Our results are stated under optimal conditions on the asymptotics of the potentials near zero and near infinity.

– It is shown that if u is a solution of the initial value problem for the generalized Korteweg–de Vries equation such that there exists b ∈ R with suppu(·, tj ) ⊆ (b,∞) (or (−∞, b)), for j = 1,2 (t1 = t2), then u≡ 0. 2002 Éditions scientifiques et médicales Elsevier SAS AMS classification: Primary 35Q53; secondary 35G25; 35D99

- Luis Vega
- 2008

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)

In this article we will study the initial value problem for some Schrödinger equations with Diraclike initial data and therefore with infinite L2 mass, obtaining positive results for subcritical nonlinearities. In the critical case and in one dimension we prove that after some renormalization the corresponding solution has finite energy. This allows us to… (More)

- Naiara Arrizabalaga, Albert Mas, Luis Vega
- SIAM J. Math. Analysis
- 2015

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)

- Benjamin Jacob Arthur, Masakazu Konishi, +11 authors Sharad Shanbhag
- 2001

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)