Talk by Luca Fanelli
On May 2nd, 2024, Luca Fanelli (Ikerbasque, Bilbao) gave a talk about "Scattering in the energy space for the 2D defocusing nonlinear Klein-Gordon equation with a magnetic field" as part of the research seminar Analysis of the FernUniversität in Hagen. This lecture is partially supported by the COST action Mathematical models for interacting dynamics on networks.
Abstract
Show Abstract
Let us consider the defocusing nonlinear Klein-Gordon equation
(1) −□um2uup = 0
in R1d where □ = −(∂tt)2∆. For p ≥ 1-4/d, the global Cauchy-Theory with scattering in H1 is due to Ginibre-Soffer-Velo (d ≥ 3) and Nakanishi (d = 1, 2). The main ingredient for the Cauchy Theory are Strichartz estimates, while for the scattering the fundamental tool is a global space-time estimate (Morawetz) given by the nonlinear term in the equation. In particular, in low dimension d = 1, 2, Nakanishi in 1999 found a way to obtain a Morawetz estimate by introducing some time-dependent multiplier. In this seminar, we will present a magnetic perturbation of equation (1) of the form
(2) −□Aum2u|u|p-1u = 0,
where
□A = −(∂t − iA0)2 Σj=1,...,d (∇ − iAj)2 . Aj = Aj(t, x) : R1d → R, j = 0, . . . , d.
For equation (2), we provide a scattering result, in the same style as in Nakanishi, in the natural Sobolev space H1A associated to the magnetic field. The main challenge is to understand the algebraic properties of the time-dependent Morawetz multipliers and how do they interplay with the magnetic field. The results are obtained in collaboration with V. Georgiev (University Pisa) and S. Lucente (University Bari).
