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.
- Pub Date:
- May 2020
- Mathematics - Analysis of PDEs
- 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