On the Cauchy Problem for the Zakharov System

We study the local Cauchy problem in time for the Zakharov system, (1.1) and (1.2), governing Langmuir turbulence, with initial data (u(0),n(0), ∂tn(0))∈Hk⊕Hlscr;⊕Hℓ−1, in arbitrary space dimensionν. We define a natural notion of criticality according to which the critical values of (k, ℓ) are (ν/2−3/2,ν/2−2). Using a method recently developed by Bourgain, we prove that the Zakharov system is locally well posed for a variety of values of (k, ℓ). The results cover the whole subcritical range forν⩾4. Forν⩽3, they cover only part of it and the lowest admissible values are (k, ℓ)=(1/2, 0) forν=2, 3 and (k, ℓ)=(0, −1/2) forν=1. As a by product of the one dimensional result, we prove well-posedness of the Benney system, (1.14) and (1.15), governing the interaction of short and long waves for the same values of (k, ℓ).