arXiv subjects
Publications and source records attributed to Fabio Pusateri.
In this manuscript we prove global existence and linear asymptotic behavior of small solutions to nonlinear wave equations. We assume that the quadratic part of the nonlinearity satisfies a non-resonant condition which is a generalization of the null condition given by Klainerman.
In 2004, Féjoz [Démonstration du 'théoréme d'Arnold' sur la stabilité du système planétaire (d'après M. Herman). Ergod. Th. & Dynam. Sys. 24(5) (2004), 1521-1582], completing investigations of Herman's [Démonstration d'un théoréme de V.I. Arnold. Séminaire de Systémes Dynamiques et manuscripts, 1998], gave a complete proof of 'Arnold's Theorem' [V. I. Arnol'd. Small denominators and problems of stability of motion in classical and celestial mechanics. Uspekhi Mat. Nauk. 18(6(114)) (1963), 91-192] on the planetary many-body problem, establishing, in particular, the existence of a positive measure set of smooth (C\infty) Lagrangian invariant tori for the planetary many-body problem. Here, using Rüßmann's 2001 KAM theory [H. Rüßmann. Invariant tori in non-degenerate nearly integrable Hamiltonian systems. R. & C. Dynamics 2(6) (2001), 119-203], we prove the above result in the real-analytic class.
We consider the free-boundary motion of two perfect incompressible fluids with different densities $ρ_+$ and $ρ_-$, separated by a surface of discontinuity along which the pressure experiences a jump proportional to the mean curvature by a factor $ε^2$. Assuming the Raileigh-Taylor sign condition and $ρ_- \leq ε^{3/2}$ we prove energy estimates uniform in $ρ_-$ and $ε$. As a consequence we obtain convergence of solutions of the interface problem to solutions of the free-boundary Euler equations in vacuum without surface tension as $ε$ and $ρ_-$ tend to zero.
We present a new proof of global existence and long range scattering, from small initial data, for the one-dimensional cubic gauge invariant nonlinear Schrödinger equation, and for Hartree equations in dimension $n \geq 2$. The proof relies on an analysis in Fourier space, related to the recent works of Germain, Masmoudi and Shatah on space-time resonances. An interesting feature of our approach is that we are able to identify the long range phase correction term through a very natural stationary phase argument.
We consider the interface problem between two incompressible and inviscid fluids in the presence of surface tension. Following the geometric approach of [Shatah,J.;Zeng,C. A priori estimates for Fluid Interface Problems. CPAM, vol.16, no.6, June 2008] we show that solutions to this problem converge to solutions of the free-boundary Euler equations in vacuum as one of the densities goes to zero.