The Zakharov system in dimension $d\ge 4$

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 Schrödinger equation at the lowest admissible regularity, global well-posedness and scattering are proved. The results cover energy-critical and energy-supercritical dimensions $d\ge 4$.