Polynomial Growth of high Sobolev Norms of solutions to the Zakharov-Kuznetsov Equation

We consider the Zakharov-Kuznetsov equation (ZK) in space dimension 2. Solutions u with initial data u_0 ∈ H s are known to be global if s ≥ 1. We prove that for any integer s ≥ 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 Schrödinger Korteweg-de-Vries equation. The main ingredients are adequate bilinear estimates in the context of Bourgain's spaces