We consider the Cauchy problem associated with the Zakharov-Kuznetsov equation, posed on T 2 . We prove the local well-posedness for given data in H s (T 2 ) whenever s > 5/3. More importantly, we prove that this equation is of quasi-linear type for initial data in any Sobolev space on the torus, in sharp contrast with its semi-linear character in the R 2 and R × T settings. Our interest here is in studying the local well-posedness issue to the IVP (1) for given data in the periodic Sobolev

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... e H s (T 2 ), s ∈ R. We note that for s > 2 one can ignore the dispersive effects and solve (1) as a quasi-linear hyperbolic equation leading to the local well-posedness of (1) in H s (T 2 ), s > 2. Our first goal is to prove a well-posedness result in spaces of lower regularity. Theorem 1.1. Let s > 5/3. Then for every w 0 ∈ H s (T 2 ) there exist a time T = T ( w 0 H s ) and a unique solution w ∈ C([0, T ]; H s (T 2 )) to the IVP (1) such that w, ∂ x w, ∂ y w ∈ L 1 T L ∞ xy . Moreover, the map that takes the initial data to the solution w 0 → w ∈ C([0, T ]; H s (T 2 )) is continuous.