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.

Partly, the notation in Section 2 is adopted from arXiv:1912.05820. v2: references fixed, further minor corrections. v3: Minor change of Theorem 4.1 and its proof, which includes low frequeny cases now. v4: fixed references to [7]

References in corpus (2)

• Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrodinger equation in the radial case
• The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions

Cited by in corpus (2)

• The three dimensional stochastic Zakharov system
• The Zakharov system in dimension $d⩾4$