The Zakharov system in dimension $d \geqslant 4$
Abstract
The sharp range of Sobolev spaces is determined in which the Cauchy problem for the classical Zakharov system is well-posed, which includes existence of solutions, uniqueness, persistence of initial regularity, and real-analytic dependence on the initial data. In addition, under a condition on the data for the Schr\"odinger equation at the lowest admissible regularity, global well-posedness and scattering is proved. The results cover energy-critical and energy-supercritical dimensions .
Keywords
Cite
@article{arxiv.1912.05820,
title = {The Zakharov system in dimension $d \geqslant 4$},
author = {Timothy Candy and Sebastian Herr and Kenji Nakanishi},
journal= {arXiv preprint arXiv:1912.05820},
year = {2024}
}
Comments
v2: typos fixed, new Section 5 on a simplified approach to LWP and small data GWP and scattering results in non-endpoint cases. v3: minor corrections in the proofs of Thm 7.6 and 7.7. v4: slight improvement of Prop. 6.1 and 6.2 to fix the proof of the uniqueness claim