The Cauchy problem for a family of two-dimensional fractional Benjamin-Ono equations

Abstract

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.

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

“…It can be proved in analogous form as it was done in [3] that w ∈ C([0, T ]; H s (T 2 )), with w, w x , w y in L 1 ([0, T ]; L ∞ xy ), and that w is the solution of the IVP (1.1). The uniqueness and the continuous dependence on the initial data also follow in a similar way as in [3].…”

“…For the two-dimensional BO equation in R 2 in [3] we established LWP in H s (R 2 ) with s > 3 2 , where the main ingredient was a Strichartz estimate, similar to that obtained by Kenig in [11]. In [3] we followed the same approach used by Linares, Pilod and Saut in [15] for dispersive perturbations of Burger's equation and in [16] for fractional Kadomtsev-Petviashvili equations.…”

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

“…This completes the proof. Also, we require the following refined Strichartz estimate, which has been proved in different contexts (see [5,27,31]). Lemma 3.2.…”

“…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.…”

The Cauchy Problem for Dissipative Benjamin-Ono equation in Weighted Sobolev spaces

“…Solutions without infinity decay in the initial data can be found in [27] and [15]. Two-dimensional versions are also of great interest in the literature, see for example [31], [8], [11], [7], [6], [12] and [36]. Recently, a higher order versions of the BO equation were studied, see [21], [35], [37], and references therein.…”