The Zakharov system in dimension $d>= 4$

Candy T, Herr S, Nakanishi K. The Zakharov system in dimension $d>= 4$. Journal of the European Mathematical Society. 2022.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 $d \geqslant 4$