GLOBAL WELL-POSEDNESS OF THE BENJAMIN–ONO EQUATION IN H¹(R)

Abstract

Abstract. We show that the Benjamin-Ono equation is globally well-posed in H s (R) for s ≥ 1. This is despite the presence of the derivative in the non-linearity, which causes the solution map to not be uniformly continuous in H s for any s [15]. The main new ingredient is to perform a global gauge transformation which almost entirely eliminates this derivative. IntroductionIn this paper we study the Cauchy problem for the Benjamin-Ono equationwhere u : R × R → R is a real-valued function, H is the spatial Hilbert transformand u 0 lives in the inhomogeneous Sobolev space H s x for some s ∈ R. Recall that these spaces can be defined via the spatial Fourier transform

On the controllability and stabilization of the linearized Benjamin-Ono equation

“…In this case, the Hilbert transform is defined via the Fourier transform as H(f ) = −i sgn(ξ) f . In addition to the previously mentioned work [1], Iório [11], Ponce [21] and more recently Koch and Tzvetkov [15], Kenig and Koenig [13] and Tao [25] have contributed with interesting local and global results for the IVP associated to (3). The best known result till now is due to Tao.…”

“…We note that the exact controllability for the nonlinear BO equation is a really interesting problem and it may be possible to use the approaches introduced in [15,25] to obtain new results in those spaces. The two-point initial-boundary value problem for this equation is out of question in its actual form (3) since the nonlocal operator is not well defined in this setting.…”

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).…”

On quadratic derivative Schrödinger equations in one space dimension

“…Then, it must be that P >j U (t)f H 1 ≤ C f >j H 1 , which should go to zero, as j → ∞ and similarly for P >j U (t)f n H 1 . The estimates in [18] are performed in the envelope spaces H 1 c , which makes the above argument rigorous.…”