Well-posedness for the Cauchy problem of the Klein-Gordon-Zakharov system in 2D

Abstract

This paper is concerned with the Cauchy problem of the 2D Zakharov-Kuznetsov equation. We prove bilinear estimates which imply local in time wellposedness in the Sobolev space H s (R 2 ) for s > −1/4, and these are optimal up to the endpoint. We utilize the nonlinear version of the classical Loomis-Whitney inequality and develop an almost orthogonal decomposition of the set of resonant frequencies. As a corollary, we obtain global well-posedness in L 2 (R 2 ). 2010 Mathematics Subject Classification. 35Q55, 35A01.

Local well-posedness for the Klein-Gordon-Zakharov system in 3D

“…In this case we need a further decomposition with respect to angular variables. Similar decompositions were also used by Bejenaru-Herr [3] and Kinoshita [6].…”

“…In fact the main result shows local well-posedness for s > − 1 2 , thus leaving open only the critical case s = − 1 2 . The proof combines the method used by Bejenaru-Herr [3] for their optimal well-posedness result for the 3D Zakharov system and Kinoshita's approach for the optimal well-posedness result for the 2D Klein-Gordon-Zakharov system [6]. Bejenaru-Herr introduced a suitable additional decomposition with respect to angular variables in frequency space, which also plays a fundamental role both in Kinoshita's article and in our paper.…”

“…In 2D Tsugawa proved local well-posedness for s ≥ − 1 2 . Also in 2D S. Kinoshita [6] substantially improved this result to s > − 3 4 . Crucial for this improvement were bilinear estimates given by Bejenaru-Herr-Holmer-Tataru [1] in order to prove an optimal local well-posedness result for the related Zakharov system in 2D based on a result by Bejenaru-Herr-Tataru [2].…”

“…It is namely easily verified that the bilinear estimates formulated in Proposition 1.1 are sufficient to cover this case. For details we refer to [6], Remarks 1 and 2.…”

Stability and instability of the standing waves for the Klein‐Gordon‐Zakharov system in one space dimension

“…The existence of the local smooth solutions to the Cauchy problem of the KGZ system can be proven by the standard Galerkin method (see, eg, Zhou‐Guo). Moreover, the global well‐posedness was established in Guo‐Yuan, Kato, Kinoshita, Merle, and Ozawa‐Tsutaya‐Tsutsumi . Let m : = −(−Δ) −1 n t with m | t =0 : = −(−Δ) −1 ν 0 , where in one space dimension.…”