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 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 ( 2 ) for s>1 2 and globally well-posed in H 1 (×𝕋) and in H s ( 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 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 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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