Bilinear Strichartz estimates for the Zakharov–Kuznetsov equation and applications

Molinet, Luc  ; Pilod, Didier 

Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015) no. 2, pp. 347-371

Résumé

This article is concerned with the Zakharov–Kuznetsov equation

$$\begin{array}{cc}{\partial }_{t}u+{\partial }_x\Delta u+u{\partial }_{x}u=0.& \text{(0.1)}\end{array}$$

We prove that the associated initial value problem is locally well-posed in $H^s\left({ℝ}^2\right)$ for $s>\frac{1}{2}$ and globally well-posed in $H^1(ℝ\times 𝕋)$ and in $H^s\left({ℝ}^3\right)$ 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 ${ℝ}^2$ 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 ${ℝ}^3$, we need to use the atomic spaces introduced by Koch and Tataru.

Détail

DOI : 10.1016/j.anihpc.2013.12.003

Classification : 35A01, 35Q53, 35Q60
Keywords: Zakharov–Kuznetsov equation, Initial value problem, Well-posedness, Bilinear Strichartz estimates, Bourgain's spaces