We obtain global well-posedness, scattering, and global L10 t,x spacetime bounds for energy-class solutions to the quintic defocusing SchrÃ¶dinger equation in R1+3, which is energy-critical. Inâ€¦ (More)

We prove an " almost conservation law " to obtain global-in-time well-posedness for the cubic, defo-cussing nonlinear SchrÃ¶dinger equation in H s (R n) when n = 2, 3 and s > 4 7 , 5 6 , respectively.

We prove that the 1D SchrÃ¶dinger equation with derivative in the nonlinear term is globally well-posed in H s , for s > 2/3 for small L 2 data. The result follows from an application of the "â€¦ (More)

We consider the cubic defocusing nonlinear SchrÃ¶dinger equation on the two dimensional torus. We exhibit smooth solutions for which the support of the conserved energy moves to higher Fourier modes.â€¦ (More)

In this paper we prove that the 1D SchrÃ¶dinger equation with derivative in the nonlinear term is globally well-posed in H s , for s > 1 2 for data small in L 2. To understand the strength of thisâ€¦ (More)

We study the long-time behaviour of the focusing cubic NLS on R in the Sobolev norms H s for 0 < s < 1. We obtain polynomial growth-type upper bounds on the H s norms, and also limit any orbital H sâ€¦ (More)

We prove an endpoint multilinear estimate for the X s,b spaces associated to the periodic Airy equation. As a consequence we obtain sharp local well-posedness results for periodic generalized KdVâ€¦ (More)

The initial value problem for the Korteweg-deVries equation on the line is shown to be globally well-posed for rough data. In particular, we show global well-posedness for initial data in H(R) forâ€¦ (More)

We prove that the modified Benjamin-Ono equation is globally wellposed in H for s â‰¥ 1/2. The exponent H seems to be optimal in the sense that the solution map is not C in H for s < 1/2 [18]. Weâ€¦ (More)

We consider the cubic defocusing nonlinear SchrÃ¶dinger equation on the two dimensional torus. We exhibit smooth solutions for which the support of the conserved energy moves to higher Fourier modes.â€¦ (More)