No 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.

In this article we prove that sufficiently smooth solutions of the Zakharov-Kuznetsov equation:that have compact support for two different times are identically zero. 2000 Mathematics Subject Classification. 35Q53, 37K05. Key words and phrases. Nonlinear dispersive equations, estimates of Carleman type. With the support of Colciencias, Colombia, Project 574-2009.

In this work we prove that the initial value problem (IVP) associated to the fractional two-dimensional Benjamin-Ono equationwhere 0 < α ≤ 1, D α x denotes the operator defined through the Fourier transform byand H denotes the Hilbert transform with respect to the variable x, is locally well posed in the Sobolev space2000 Mathematics Subject Classification. 35Q53, 37K05. Key words and phrases. Benjamin Ono equation. 1 in the less dispersive case (α ∈ (0, 1)), and for s > 3/2 − 3α/8 they proved the LWP of the Cauchy problem for the equation (1.6) in the Sobolev space H s (R). Using the same approach as that used in [17], Linares, Pilod and Saut in [18] considered the family of fractional Kadomtsev-Petviashvili equations7) and established the LWP in Y s of the IVP associated to the equations (1.7), with s > 2 − α/4. The main ingredient in the well-posedness analysis in the works [17] and [18] is a Strichartz estimate, similar to that obtained by Kenig and Koenig in [10] and by Kenig in [9].Inspired by the works [17] and [18], in this paper we also study the relation between the amount of dispersion and the size of the Sobolev space in order to have LWP of the family of IVPs (1.1).Following the scheme developed in [18], we also obtain a Strichartz estimate (see Lemma 3.4 below), which, combined with some energy estimates, allows us to prove the LWP of the IVP (1.

No abstract