Bilinear Strichartz estimates for the Zakharov–Kuznetsov equation and applications
Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015) no. 2, pp. 347-371.

This article is concerned with the Zakharov–Kuznetsov equation

∂ t u+∂ x Δu+u∂ x u=0.(0.1)
We prove that the associated initial value problem is locally well-posed in H s (ℝ 2 ) for s>1 2 and globally well-posed in H 1 (ℝ×𝕋) and in H s (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 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 : 10.1016/j.anihpc.2013.12.003
Classification : 35A01, 35Q53, 35Q60
Mots-clés : Zakharov–Kuznetsov equation, Initial value problem, Well-posedness, Bilinear Strichartz estimates, Bourgain's spaces 
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     author = {Molinet, Luc and Pilod, Didier},
     title = {Bilinear {Strichartz} estimates for the {Zakharov{\textendash}Kuznetsov} equation and applications},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {347--371},
     publisher = {Elsevier},
     volume = {32},
     number = {2},
     year = {2015},
     doi = {10.1016/j.anihpc.2013.12.003},
     mrnumber = {3325241},
     zbl = {1320.35106},
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Molinet, Luc; Pilod, Didier. Bilinear Strichartz estimates for the Zakharov–Kuznetsov equation and applications. Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015) no. 2, pp. 347-371. doi : 10.1016/j.anihpc.2013.12.003. http://www.numdam.org/articles/10.1016/j.anihpc.2013.12.003/

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