We consider the Schrödinger map initial-value problemwhere φ :we prove that the Schrödinger map initial-value problem admits a unique global smooth solution φ ∈ C(R :Q is smooth and satisfies the smallness condition φ0 −Q Ḣd/2 1. We prove also that the solution operator extends continuously to the space of data inḢ

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.

Abstract. We prove that the KP-I initial-value problemis globally well-posed in the energy spacex ∂ y φ L 2 < ∞}.

We consider the gravity water waves system in the case of a one dimensional interface, for sufficiently smooth and localized initial data, and prove global existence of small solutions. This improves the almost global existence result of Wu [Wu09]. We also prove that the asymptotic behavior of solutions as time goes to infinity is different from linear, unlike the three dimensional case [GMS12a,Wu11]. In particular, we identify a suitable nonlinear logarithmic correction and show modified scattering. The solutions we construct in this paper appear to be the first global smooth nontrivial solutions of the gravity water waves system in 2d.

Following the program started in [24], we attempt to remove the analyticity assumption in the the well known Hawking-Carter-Robinson uniqueness result for regular stationary vacuum black holes. Unlike [24], which was based on a tensorial characterization of the Kerr solutions, due to Mars [29], we rely here on Hawking's original strategy, which is to reduce the case of general stationary space-times to that of stationary and axi-symmetric spacetimes for which the Carter-Robinson uniqueness result holds. In this reduction Hawking had to appeal to analyticity. Using a variant of the geometric Carleman estimates developed in [24], in this paper we show how to bypass analyticity in the case when the stationary vacuum space-time is a small perturbation of a given Kerr solution. Our perturbation assumption is expressed as a uniform smallness condition on the Mars-Simon tensor. The starting point of our proof is the new local rigidity theorem established in [2].