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- J. COLLIANDER
- 2001

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 2-based Sobolev spaces H s where local well-posedness is presently known, apart from the H 1 4 (R) endpoint for mKdV. The result… (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)

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)

- J Colliander, M Keel, G Staffilani, H Takaoka, T Tao
- 2004

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)

- J. Colliander, Markus Keel, Gigliola Staffilani, Hideo Takaoka, Terence Tao
- SIAM J. Math. Analysis
- 2001

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)

- J Colliander, M Keel, G Staffilani, H Takaoka, T Tao
- 2002

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 KdV equations, as well as some global well-posedness results below the energy norm.

- J. COLLIANDER
- 2010

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 u 0 with u 0 H s < δ, so that the… (More)

- J. Colliander, Markus Keel, Gigliola Staffilani, Hideo Takaoka, Terence Tao
- SIAM J. Math. Analysis
- 2002

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)

- J Colliander, M Keel, G Staffilani, H Takaoka, T Tao
- 2008

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