Bilinear Strichartz estimates for the Zakharov–Kuznetsov equation and applications

Source abstract

This article is concerned with the Zakharov–Kuznetsov equation \partial _{t}u + \partial _{x}\mathrm{\Delta }u + u\partial _{x}u = 0. We prove that the associated initial value problem is locally well-posed in H^{s}(\mathbb{R}^{2}) for s > \frac{1}{2} and globally well-posed in H^{1}(\mathbb{R} \times \mathbb{T}) and in H^{s}(\mathbb{R}^{3}) for s > 1 . Our main new ingredient is a bilinear Strichartz estimate in the context of Bourgain's spaces which allows to control the high-low frequency interactions appearing in the nonlinearity of (0.1). In the \mathbb{R}^{2} case, we also need to use a recent result by Carbery, Kenig and Ziesler on sharp Strichartz estimates for homogeneous dispersive operators. Finally, to prove the global well-posedness result in \mathbb{R}^{3} , we need to use the atomic spaces introduced by Koch and Tataru.