Quadratic Morawetz inequalities and asymptotic completeness in the energy space for nonlinear Schr"odinger and Hartree equations

TL;DR

This paper reviews recent advances in quadratic Morawetz inequalities, demonstrating their role in proving asymptotic completeness for nonlinear Schrödinger and Hartree equations in various dimensions.

Contribution

It provides a pedagogical overview of quadratic Morawetz inequalities and their application to asymptotic completeness in the energy space for these equations.

Findings

Quadratic Morawetz inequalities facilitate simpler proofs of asymptotic completeness.

The results apply to nonlinear Schrödinger equations in any space dimension.

The methods extend to Hartree equations in dimensions greater than two in noncritical cases.

Abstract

Recently several authors have developed multilinear and in particular quadratic extensions of the classical Morawetz inequality. Those extensions provide (among other results) an easy proof of asymptotic completeness in the energy space for nonlinear Schr"odinger equations in arbitrary space dimension and for Hartree equations in space dimension greater than two in the noncritical cases. We give a pedagogical review of the latter results.