The NLS approximation for two dimensional deep gravity waves

This article is concerned with infinite depth gravity water waves in two space dimensions. We consider this system expressed in position-velocity potential holomorphic coordinates. Our goal is to study this problem with small wave packet data, and to show that this is well approximated by the cubic nonlinear Schr\"odinger equation (NLS) on the natural cubic time scale.Comment: 23 page

Global bounds for the cubic nonlinear Schr\"odinger equation (NLS) in one space dimension

This article is concerned with the small data problem for the cubic nonlinear Schr\"odinger equation (NLS) in one space dimension, and short range modifications of it. We provide a new, simpler approach in order to prove that global solutions exist for data which is small in $H^{0,1}$. In the same setting we also discuss the related problems of obtaining a modified scattering expansion for the solution, as well as asymptotic completeness.Comment: 15 pages. We fixed the proof of Lemma 2.

Two dimensional water waves in holomorphic coordinates II: global solutions

This article is concerned with the infinite depth water wave equation in two space dimensions. We consider this problem expressed in position-velocity potential holomorphic coordinates,and prove that small localized data leads to global solutions. This article is a continuation of authors' earlier paper arXiv:1401.1252.Comment: 21 pages. We have updated the authors' inf

The lifespan of small data solutions in two dimensional capillary water waves

This article is concerned with the incompressible, irrotational infinite depth water wave equation in two space dimensions, without gravity but with surface tension. We consider this problem expressed in position-velocity potential holomorphic coordinates,and prove that small data solutions have at least cubic lifespan while small localized data leads to global solutions.Comment: Typos corrected, references updated, final versio

Two dimensional gravity water waves with constant vorticity: I. Cubic lifespan

This article is concerned with the incompressible, infinite depth water wave equation in two space dimensions, with gravity and constant vorticity but with no surface tension. We consider this problem expressed in position-velocity potential holomorphic coordinates, and prove local well-posedness for large data, as well as cubic lifespan bounds for small data solutions.Comment: 64 page

Enhanced Lifespan of Smooth Solutions of a Burgers-Hilbert Equation

We consider an initial value problem for a quadratically nonlinear inviscid Burgers-Hilbert equation that models the motion of vorticity discontinuities. We use a normal form transformation, which is implemented by means of a near-identity coordinate change of the independent spatial variable, to prove the existence of small, smooth solutions over cubically nonlinear time-scales. For vorticity discontinuities, this result means that there is a cubically nonlinear time-scale before the onset of filamentation.Comment: 13 pages, 1 figur

CAN SOCIAL PROTECTION REALLY MAKE A SIGNIFICANT CONTRIBUTION TO POVERTY REDUCTION? THE CASE OF ROMANIA

Most Romanians believe that the state should assume more responsibility for the welfare of everyone. Social protection must actually be understood in the broader framework of gradual and more alert transfer of the responsibility from the individual to the state level. If in the case of a minimal state the individuals would be forced to save to cope with unforeseen situations like job loss, disability or illness, in the case of a welfare state, which guarantees minimum incomes, these reasons fade. Individuals have increasing expectations from the authorities, and largely decline their capabilities of helping others through charity or philanthropy. In the light of the lack of confidence in the strength of private actions to support those in need, public solutions are expected to eliminate poverty through social protection programs. The purpose of this paper is to analyze the ability of social protection programs in Romania to help improve well-being among the most disadvantaged citizens of Romania and the costs associated with such objective

THE EUROPEAN BUSINESS CYCLE

The construction of European Monetary Union has raised several questions about the existence of a common business cycle, a European one. The lack of cyclical synchronization would complicate the monetary and fiscal policies in the Union, being a negativeEuropean business cycle, correlation, synchronization of business cycles

Finite depth gravity water waves in holomorphic coordinates

In this article we consider irrotational gravity water waves with finite bottom. Our goal is two-fold. First, we represent the equations in holomorphic coordinates and discuss the local well-posedness of the problem in this context. Second, we consider the small data problem and establish cubic lifespan bounds for the solutions. Our results are uniform in the infinite depth limit, and match the earlier infinite depth result of Hunter-Ifrim-Tataru.Comment: 82 pages, 1 figur