Periodic Cauchy problem for one two-dimensional generalization of the Benjamin–Ono equation in Sobolev spaces of low regularity

Abstract

In this work we prove that the initial value problem (IVP) associated to the two-dimensional Benjamin-Ono equationwhere H denotes the Hilbert transform with respect to the variable x and ∆ is the Laplacian with respect to the spatial variables x and y, is locally well-posed in the periodic Sobolev space H s (T 2 ), with s > 7/4. Using the abstract theory developed by Kato in [9] and [10], it can be established the local well-posedness (LWP) of the IVP (1.1) in H s (T 2 ), with s > 2. Nevertheless this approach ignores the dispersive effects of the linear part of the equation in (1.1). The Cauchy problem for the one-dimensional BO equation (1.4) has been extensively studied on the real line and in the periodic setting. On the real line, using the dispersive character of the linear part of the equation, global well-posedness (GWP) of the IVP for the BO equation (1.4) has been established in H s (R), for s = 3 2 by Ponce in [22] and LWP was proved for s > 5 4 in [13] by Koch and Tzvetkov. In [12], based on ideas of Koch and Tzvetkov in [13], Kenig and Koenig obtained a refined version of the Strichartz estimate, which allowed them to establish LWP of the 2000 Mathematics Subject Classification. 35Q53.

The IVP for a certain dispersion generalized ZK equation in bi-periodic spaces

“…Therefore, we use the short-time Strichartz linear approach introduced by Koch and Tzvetkov in [21], to get local well-posedness of Benjamin-Ono (BO) equation in R, which has been proved to be useful for these types of equations (see [14,26,17,18]). But to perform this task in two dimensions with periodic context, we adapt the method used to prove local wellposedness for the Cauchy problem associated to the third-order KP-I and fifth-order KP-I equations on R × T and T 2 proposed by Ionescu and Kenig in [15] (for other applications, see [25,4] and the references therein). First, a localized Strichartz-type estimate for the linear part of the equation is obtained, where the main difficulty lies in obtaining bounds for exponential sums in the periodic case (see [13]), such sums have been treated in different contexts in number theory.…”

Well-posedness for a two-dimensional dispersive model arising from capillary-gravity flows

“…Concerning the IVP (1.2), by adapting the short-time linear Strichartz estimate approach employed in [27,31], LWP in H s (R 2 ) s > 3/2 was deduced in [5]. In [4], inspired by the works in [20,30], LWP was established in H s (T 2 ) s > 7/4 assuming that the initial data satisfy 2π 0 u 0 (x, y) dx = 0 for almost every y. Recently, in [44], by employing short-time bilinear Strichartz estimates the conclusion on the periodic setting was improved to regularity s > 3/2 without any assumption on the initial data.…”