Global wellposedness for the energy-critical Zakharov system below the ground state

TL;DR

This paper proves global well-posedness for the energy-critical Zakharov system in four dimensions for initial data below the ground state, using new Strichartz and bilinear Fourier restriction estimates.

Contribution

It introduces novel Strichartz and bilinear Fourier restriction estimates for the Schrödinger equation with potential, enabling analysis below the ground state threshold.

Findings

Global well-posedness for the Zakharov system in 4D energy space

Strichartz estimates uniform for potentials below ground state

Bilinear Fourier restriction estimate for inhomogeneous Schrödinger solutions

Abstract

The Cauchy problem for the Zakharov system in the energy-critical dimension $d=4$ is considered. We prove that global well-posedness holds in the full (non-radial) energy space for any initial data with energy and wave mass below the ground state threshold. The result is based on a Strichartz estimate for the Schr\"odinger equation with a potential. More precisely, a Strichartz estimate is proved to hold uniformly for any potential solving the free wave equation with mass below the ground state constraint. The key new ingredient is a bilinear (adjoint) Fourier restriction estimate for solutions of the inhomogeneous Schr\"odinger equation with forcing in dual endpoint Strichartz spaces.