J. Colliander

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We prove global existence and scattering for the defocusing, cubic nonlinear Schrödinger equation in Hs(R3) for s > 4 5 . The main new estimate in the argument is a Morawetz-type inequality for the solution φ. This estimate bounds ‖φ(x, L4x,t(R3×R) , whereas the well-known Morawetz-type estimate of Lin-Strauss controls ∫ ∞ 0 ∫ R3 (φ(x,t)) |x| dxdt.
We prove that the 1D Schrödinger equation with derivative in the nonlinear term is globally well-posed in H, for s > 2/3 for small L data. The result follows from an application of the “I-method”. This method allows to define a modification of the energy norm H that is “almost conserved” and can be used to perform an iteration argument. We also remark that(More)
In this paper we prove that the 1D Schrödinger equation with derivative in the nonlinear term is globally well-posed in H, for s > 1 2 for data small in L. To understand the strength of this result one should recall that for s < 1 2 the Cauchy problem is ill-posed, in the sense that uniform continuity with respect to the initial data fails. The result(More)
We consider the cubic defocusing nonlinear Schrödinger equation on the two dimensional torus. We exhibit smooth solutions for which the support of the conserved energy moves to higher Fourier modes. This behavior is quantified by the growth of higher Sobolev norms: given any δ 1,K 1, s > 1, we construct smooth initial data u0 with ‖u0‖Hs < δ, so that the(More)
We prove low-regularity global well-posedness for the 1d Zakharov system and 3d Klein-Gordon-Schrödinger system, which are systems in two variables u : Rx × Rt → C and n : Rx × Rt → R. The Zakharov system is known to be locally well-posed in (u, n) ∈ L2×H−1/2 and the Klein-Gordon-Schrödinger system is known to be locally well-posed in (u, n) ∈ L × L. Here,(More)