On special regularity properties of solutions of the benjamin-ono-zakharov-kuznetsov (bo-zk) equation

Abstract

In this paper we study special properties of solutions of the initial value problem (IVP) associated to the Benjamin-Ono-Zakharov-Kuznetsov (BO-ZK) equation. We prove that if initial data has some prescribed regularity on the right hand side of the real line, then this regularity is propagated with infinite speed by the flow solution. In other words, the extra regularity on the data propagates in the solutions in the direction of the dispersion. The method of proof to obtain our result uses weighted energy estimates arguments combined with the smoothing properties of the solutions. Hence we need to have local well-posedness for the associated IVP via compactness method. In particular, we establish a local well-posedness in the usual L 2 (R 2)-based Sobolev spaces H s (R 2) for s > 5 4 which coincides with the best available result in the literature proved employing more complicated tools.

On some regularity properties for the dispersive generalized Benjamin-Ono-Zakharov-Kuznetsov Equation

“…We shall emphasize that the Sobolev regularity attained in Theorem 1.1 does not yield to an improvement with respect to the conclusions in [34], and to the results in [7,30] for α = 0. Nevertheless, when 0 ≤ α < 1, Theorem 1.1 states the bestknown result involving solutions of (1.1) in the class (1.7).…”

“…Remark 1.3. In the case of physical relevance α = 0 in (1.1), i.e., the BOZK equation, Theorem 1.2 leads to an extension to the fractional setting of the conclusions derived in [30,Theorem 1.4] concerning the propagation of regularity principle for local derivatives.…”

“…In particular, in the case of the BOZK equation, α = 0 in (1.1), the LWP result in [30,Theorem 1.3] determines the validity of Theorem 1.2 for initial data with regularity H r (R 2 ), r > 5/4. Remark 1.5.…”

On decay properties for solutions of the Zakharov-Kuznetsov equation

“…They proved that extra regularity in the initial data localized on the right-hand side of the real line travels to the left with an infinite speed. Since this pioneering work, the study of the propagation of regularity has been investigated for other dispersive equations: In dimension n = 1, see [20,25,26,31,41,45,47,52,62], and in higher dimensions, n ≥ 2, see [18,27,43,47,54,55,56]. For a more recent survey about the study of the propagation of the regularity principle, we refer to [44].…”

Persistence properties for the dispersion generalized BO-ZK equation in weighted anisotropic Sobolev spaces

“…These results were shown to be sharp in the sense that a sufficiently smooth nontrivial solution do not persist in L 2 ((1 + x 2 + y 2 ) 7/2 dxdy). For recent results concerning local well-posedness in the standard Sobolev spaces we refer the reader to [3] and [25].…”