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On the Cauchy problem for the Zakharov system. (English) Zbl 0894.35108
The authors study the local Cauchy problem in time for the Zakharov system in arbitrary space dimension $$D$$. It is used a contraction or fixed point method, which is a simpler version than that used by Bourgain. A basic tool is the extension to the Zakharov system the notion of criticality which applies to dilation covariant equations as the nonlinear Schrödinger equation. In the Zakharov system, the two equations have dilation invariance but the two relevant dilation transformations are incompatible.
The results cover the whole subcritical range for $$d\geq 4$$, while for $$d\leq 3$$, they cover only part of it.