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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, 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.

- J. COLLIANDER, H. TAKAOKA
- 2003

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.