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arXiv · 2005.09932

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

Timothy Candy·Sebastian Herr·Kenji Nakanishi
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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.

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Timothy Candy, Sebastian Herr, Kenji Nakanishi. 2020-05-20. Global wellposedness for the energy-critical Zakharov system below the ground state. https://doi.org/10.1016/j.aim.2021.107746

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