J. Colliander

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We obtain global well-posedness, scattering, and global L 10 t,x space-time bounds for energy-class solutions to the quintic defocusing Schrödinger equation in R 1+3 , which is energy-critical. In particular, this establishes global existence of classical solutions. Our work extends the results of Bourgain [4] and Grillakis [20], which handled the radial(More)
In a recent paper [18], Kenig, Ponce and Vega study the low regularity behavior of the focusing nonlinear Schrödinger (NLS), focusing modified Korteweg-de Vries (mKdV), and complex Korteweg-de Vries (KdV) equations. Using soliton and breather solutions, they demonstrate the lack of local well-posedness for these equations below their respective endpoint(More)
We obtain global well-posedness, scattering, and global L 10 t,x space-time bounds for energy-class solutions to the quintic defocusing Schrödinger equation in R 1+3 , which is energy-critical. In particular, this establishes global existence of classical solutions. Our work extends the results of Bourgain [4] and Grillakis [20], which handled the radial(More)
We prove that the 1D Schrödinger 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 " I-method ". This method allows to define a modification of the energy norm H 1 that is " almost conserved " and can be used to perform an iteration argument. We also(More)
The nonlinear wave and Schrödinger equations on R d , with general power non-linearity and with both the focusing and defocusing signs, are proved to be ill-posed in the Sobolev space H s whenever the exponent s is lower than that predicted by scaling or Galilean invariances, or when the regularity is too low to support distributional solutions. This(More)
In this paper we prove that the 1D Schrödinger 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 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 prove global existence and scattering for the defocusing, cubic nonlinear Schrödinger equation in H s (R 3) 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, t) L 4 x,t (R 3 ×R) , whereas the well-known Morawetz-type estimate of Lin-Strauss controls ∞ 0 R 3 (φ(x,t)) 4 |x| dxdt.