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

Abstract

We prove that the initial value problem associated to a nonlocal perturbation of the Benjamin-Ono equation is locally and globally well-posed in Sobolev spaces H s (R) for any s > −3/2 and we establish that our result is sharp in the sense that the flow map of this equation fails to be C 2 in H s (R) for s < −3/2. Finally, we study persistence properties of the solution flow in the weighted Sobolev spaces Zs,r = H s (R) ∩ L 2 (|x| 2r dx) for s ≥ r > 0. We also prove some unique continuation properties of the solution flow in these spaces.

Well-posedness for Fractional Growth-Dissipative Benjamin-Ono Equations

“…Regarding the IVP for the npBO equation (1.3), we obtain the same GWP in H s (R) established in [8] and we deduce new global results in periodic Sobolev spaces. More specifically, we deduce GWP in H s (R) and H s (T) for s > −3/2 and sharp results in the sense that the flow map u 0 → u for npBO fails to be C 2 at zero from H s (R) to H s (R) or from H s (T) to H s (T) when s < −3/2.…”

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

“…In the following we will obtain the global well-posedness. First we note that the case a = 1 can be approach by using the ideas in [14]. Thus, we lead only with the case a ∈ (0, 1).…”

“…The proof of the Theorems 1.2 and 1.3 in the case a = 1 can be obtained by the same approach of [14]. Thus, in the following we deal only with the case a ∈ (0, 1).…”

“…In fact, we made use of an additional commutator estimate for the derivative D γ ξ (see Proposition 2.12 below), see also [17] and [7]. The proof of Theorem 1.1 will be obtained by using the ideas of Fonseca, Pastrán, and Rodríguez-blanco [14], as well as the Stein-Weiss interpolation theorem with change of measure, see [3]. In this work the authors studied a version of the BO equation with a dissipative effect.…”