# Bilinear Strichartz estimates for the ZK equation and applications

Abstract : 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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Journal articles

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https://hal.archives-ouvertes.fr/hal-01205993
Contributor : Luc Molinet <>
Submitted on : Monday, September 28, 2015 - 11:55:58 AM
Last modification on : Sunday, March 29, 2020 - 6:24:03 PM
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LWP ZK (final 26-03-13).pdf
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• HAL Id : hal-01205993, version 1

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

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