Global well-posedness of the Benjamin–Ono equation in low-regularity spaces

Abstract

We prove that the Benjamin–Ono initial-value problem is globally well-posed in the Banach spaces H r σ ( R ) H^\sigma _r(\mathbb {R}) , σ ≥ 0 \sigma \geq 0 , of real-valued Sobolev functions.

Ill-posedness results for the (generalized) Benjamin-Ono-Zakharov-Kuznetsov equation

“…It seems not to be easy to obtain a reasonable improvement of Theorem 1.1. We observe that recently Tao [28], Burq and Planchon [7], and Ionescu and Kenig [13] have obtained much stronger results for the Cauchy problem associated with the BO equation. However, it should be pointed out that their results are established by constructing appropriate gauge transformations.…”

Well-posedness for the nonlinear fractional Schrödinger equation and inviscid limit behavior of solution for the fractional Ginzburg-Landau equation

“…Then, we give definitions of some dyadic spaces, the properties of which can be found in [5]. First, we define the dyadic decomposition.…”

Stability in H1/2 of the sum of K solitons for the Benjamin–Ono equation

“…The existence of global weak solutions u ∈ C([0, ∞); H 1/2 (R)) ∩ C 1 ((0, ∞); H −3/2 (R)) to (1.2) with energy space initial data 1 The Fourier transform is given bŷ u(0, x) = u 0 (x) ∈ H 1/2 (R) was shown by J. C. Saut [19] (see also the paper of J. Ginibre and G. Velo [7]). For the strong H s -solution, A. Ionescu and C. E. Kenig [8] established global well-posedness for s ≥ 0 (see also the paper of T. Tao [20]). This solution conserves the functional N (u) (and E(u) when s ≥ 1/2).…”