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

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 Schrödinger 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 Schrödinger equation with forcing in dual endpoint Strichartz spaces.