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Journals and Conferences
The initial value problems for the Korteweg-de Vries (KdV) and modified KdV (mKdV) equations under periodic and decaying boundary conditions are considered. These initial value problems are shown to be globally well-posed in all L-based Sobolev spaces Hs where local well-posedness is presently known, apart from the H 1 4 (R) endpoint for mKdV. The result… (More)
We prove an “almost conservation law” to obtain global-in-time well-posedness for the cubic, defocussing nonlinear Schrödinger equation in H(R) when n = 2, 3 and s > 4 7 , 5 6 , respectively.
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 an endpoint multilinear estimate for the X spaces associated to the periodic Airy equation. As a consequence we obtain sharp local well-posedness results for periodic generalized KdV equations, as well as some global well-posedness results below the energy norm.
We consider the Cauchy problem for a family of semilinear defocusing Schrödinger equations with monomial nonlinearities in one space dimension. We establish global well-posedness and scattering. Our analysis is based on a four-particle interaction Morawetz estimate giving a priori Lt,x spacetime control on solutions.
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)
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 Hs(R) for − 3 10 < s.