Fabio Pusateri

Scattering for the Zakharov System in 3 Dimensions

We prove global existence and scattering for small localized solutions of the Cauchy problem for the Zakharov system in 3 space dimensions. The wave component is shown to decay pointwise at the optimal rate of t−1, whereas the Schrödinger component decays almost at a rate of t−7/6.

Analytic Lagrangian tori for the planetary many-body problem

In 2004 J. Fejoz [7], completing investigations of M. Herman’s [9], gave a complete proof of “Arnold’s Theorem” [1] on the planetary many‐body problem, establishing, in particular, the existence of a positive measure set of smooth (C 1 ) Lagrangian invariant tori for the planetary many‐body problem. Here, using Rusmann’s 2001 KAM theory [16], we prove the above result in the real‐analytic class.

Modified Scattering for the Boson Star Equation

We consider the question of scattering for the boson star equation in three space dimensions. This is a semi-relativistic Klein-Gordon equation with a cubic nonlinearity of Hartree type. We combine weighted estimates, obtained by exploiting a special null structure present in the equation, and a refined asymptotic analysis performed in Fourier space, to obtain global solutions evolving from small and localized Cauchy data. We describe the behavior of such solutions at infinity by identifying a suitable nonlinear asymptotic correction to scattering. As a byproduct of the weighted energy estimates alone, we also obtain global existence and (linear) scattering for solutions of semi-relativistic Hartree equations with potentials decaying faster than Coulomb.

On the limit as the surface tension and density ratio tend to zero for the two-phase euler equations

We consider the free-boundary motion of two perfect incompressible fluids with different densities

Recent advances on the global regularity for irrotational water waves

\rho_+

Almost global existence for cubic nonlinear Schrödinger equations in one space dimension

and

On global solutions of a Zakharov type system

\rho_-

Global solutions for the gravity water waves system in 2d

, separated by a surface of discontinuity along which the pressure experiences a jump proportional to the mean curvature by a factor

Nonlinear fractional Schrödinger equations in one dimension

\epsilon^2

A new proof of long-range scattering for critical nonlinear Schrödinger equations

. Assuming the Raileigh-Taylor sign condition and