Global Well-posedness of the Benjamin–ono Equation in Low-regularity Spaces

whereH is the Hilbert transform operator defined (on the spaces C(R : H), σ ∈ R) by the Fourier multiplier −i sgn(ξ). The Benjamin–Ono equation is a model for one-dimensional long waves in deep stratified fluids ([1] and [16]) and is completely integrable. The initial-value problem for this equation has been studied extensively for data in the Sobolev spaces H r (R), σ ≥ 0. It is known that the Benjamin–Ono initial-value problem has weak solutions in H r (R), H 1/2 r (R), and H r (R) (see [5], [25], and [18]) and is globally well-posed in H r (R), σ ≥ 1 (see [22], as well as [7], [17], [12], and [8] for earlier local and global well-posedness results in higher regularity spaces). In this paper we prove that the Benjamin–Ono initial-value problem is globally well-posed in H r (R), σ ≥ 0. Let H∞ r (R) = ⋂∞ σ=0 H σ r (R) with the induced metric. Let S∞ : H∞ r (R) → C(R : H∞ r (R)) denote the (nonlinear) mapping that associates to any data φ ∈ H∞ r the corresponding classical solution u ∈ C(R : H∞ r ) of the initial-value problem (1.1). We will use the L conservation law: if φ ∈ H∞ r and u = S∞(φ), then