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 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.
Keywords
Cite
@article{arxiv.2005.09932,
title = {Global wellposedness for the energy-critical Zakharov system below the ground state},
author = {Timothy Candy and Sebastian Herr and Kenji Nakanishi},
journal= {arXiv preprint arXiv:2005.09932},
year = {2023}
}
Comments
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]