The Zakharov–Kuznetsov equation in weighted Sobolev spaces

Abstract

Abstract. In this work we consider the initial value problem (IVP) associated to the two dimensional Zakharov-Kuznetsov equationpx, yq P R 2 , t P R, upx, y, 0q " u 0 px, yq. *We study the well-posedness of the IVP in the weighted Sobolev spaceswith s, r P R.

The IVP for a nonlocal perturbation of the Benjamin-Ono equation in classical and weighted Sobolev spaces

“…Fonseca, Linares and Ponce obtained, with the same techniques, similar results for the dispersion generalized Benjamin-Ono equation in [12]. For results regarding well-posenedness in these weighted spaces for other dispersive equations as gKdV, Zakharov-Kuznetsov, Benjamin, and Schrödinger see [13], [3] and [14], [20], [26], respectively. Remark 1.2.…”

Initial-boundary value problems in a half-strip for two-dimensional Zakharov–Kuznetsov equation

“…The theory of solubility and well-posedness for ZK equation and its generalizations is most developed for the pure initial-value problem. For the considered two-dimensional case the corresponding results in different functional spaces can be found in [32,6,7,2,26,27,31,15,3,18,30,16,17]. For initial-boundary value problems the theory is most developed for domains of a type I × R , where I is an interval (bounded or unbounded) on the variable x , that is, the variable y varies in the whole line ( [8,9,11,10,33,12,5]).…”

Dispersive Blow-up for Solutions of the Zakharov-Kuznetsov equation

“…Next we present the smoothing properties of the solutions of the linear problem (1.4). We start by recalling the dispersive estimates [6] and [25]). This inequalities imply the Strichartz estimates for the linear propagator.…”

“…Inequality (2.3) was originally proved for solutions of the linear problem associated with equation (1.1). However, the proof of the lemma follows the same strategy (see also Section 2 in [6]).…”

“…x norm we must introduce some weights which bring into play the local wellposedness theory for the IVP (1.2) in weighted Sobolev spaces. Recently, Fonseca and Pachon [11] (see also [6]) established the theory in these spaces. Then our next concern is to estimate D 1 +…”