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Article Dans Une Revue Annales de l'Institut Henri Poincaré C, Analyse non linéaire Année : 2015

Bilinear Strichartz estimates for the ZK equation and applications

Résumé

We prove that the associated initial value problem is locally well-posed in $H^s(\mathbb R^2)$ for $s>\frac12$ and globally well-posed in $H^1(\mathbb R\times \mathbb T)$ 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 \eqref{ZK0}. 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 $ \R^3 $, we need to use the atomic spaces introduced by Koch and Tataru.
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Dates et versions

hal-01205993 , version 1 (28-09-2015)

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  • HAL Id : hal-01205993 , version 1

Citer

Luc Molinet, Didier Pilod. Bilinear Strichartz estimates for the ZK equation and applications. Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 2015, 32 (2), pp.347-371. ⟨hal-01205993⟩
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