  We consider the Zakharov-Kuznetsov equation (ZK) in space dimension 2.
Solutions u with initial data u\_0 $\in$ H s are known to be global if s $\ge$
1. We prove that for any integer s $\ge$ 2, u(t) H s grows at most polynomially
in t for large times t. This result is related to wave turbulence and how a
solution of (ZK) can move energy to high frequencies. It is inspired by
analoguous results by Staffilani on the non linear Schr{\"o}dinger
Korteweg-de-Vries equation. The main ingredients are adequate bilinear
estimates in the context of Bourgain's spaces.
