On the Cauchy problem for the Zakharov system

Abstract

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). © 1997 Academic Press.