The Zakharov system in dimension $d⩾4$

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 is proved. The results cover energy-critical and energy-supercritical dimensions $d⩾4$.

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

References in corpus (2)

• Convolutions of singular measures and applications to the Zakharov system
• Global wellposedness for the energy-critical Zakharov system below the ground state