Bilinear Strichartz estimates for the Zakharov–Kuznetsov equation and applications
Annales de l'I.H.P. Analyse non linéaire, Volume 32 (2015) no. 2, p. 347-371
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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 ${H}^{1}\left(ℝ×𝕋\right)$ 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.

DOI : https://doi.org/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
@article{AIHPC_2015__32_2_347_0,
author = {Molinet, Luc and Pilod, Didier},
title = {Bilinear Strichartz estimates for the Zakharov--Kuznetsov equation and applications},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
publisher = {Elsevier},
volume = {32},
number = {2},
year = {2015},
pages = {347-371},
doi = {10.1016/j.anihpc.2013.12.003},
zbl = {1320.35106},
mrnumber = {3325241},
language = {en},
url = {http://www.numdam.org/item/AIHPC_2015__32_2_347_0}
}

Molinet, Luc; Pilod, Didier. Bilinear Strichartz estimates for the Zakharov–Kuznetsov equation and applications. Annales de l'I.H.P. Analyse non linéaire, Volume 32 (2015) no. 2, pp. 347-371. doi : 10.1016/j.anihpc.2013.12.003. http://www.numdam.org/item/AIHPC_2015__32_2_347_0/

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