Well-Posedness for the Two-Dimensional Zakharov-Kuznetsov Equation

Abstract

Considering the initial value problem for the two-dimensional Zakharov-Kuznetsov equation

u_t + ∂_xΔ_u + ∂_x u² = 0,

we will show its global-in-time well-posedness in H^s(R²) when 11/13 < s < 1 via the I-method by Colliander, Keel, Staffilani, Takaoka and Tao. Also we need to use Strichartz estimates and local well-posedness results for 1/2 < s. Additionally, local well-posedness for the symmetrized ZK equation in B_2,1^1/2(R²) is established in the context of atomic spaces introduced by Koch and Tataru.