Almost conservation laws and global rough solutions to a Nonlinear Schr

By James E. Colliander, Markus Keel, Gigliola Staffilani, Hideo Takaoka, T. Tao. 2002

255 citations recorded by the source corpus.

We prove an “almost conservation law” to obtain global-in-time well-posedness for the cubic, defocussing nonlinear Schrodinger equation in Hs(R) when n = 2, 3 and s > 4 7 , 5 6 , respectively.

Subfields: Control and Systems Engineering, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/b4c77ef4dd90691f4bec004500fa6ae50f90ef50

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In Control and Systems Engineering

In Mathematical Physics

In Statistical and Nonlinear Physics

Citations in the corpus

Minimal-mass blowup solutions of the mass-critical NLS (2006, methodology): “There has been some progress in lowering the regularity of H 1 x ðRÞ for these results, see [4], [9], [11], [13], [14], [17], [43], but these methods are unlikely to reach all the way to the scale-invariant regularity L(2) xðRdÞ.”
The cubic nonlinear Schr\"odinger equation in two dimensions with radial data (2007, background): “There has been much work [5, 14, 15, 17, 18, 19, 20, 26, 73, 76] devoted to lowering the regularity of the initial data from H 1 x toward L 2 x(R) and thus, toward establishing the conjecture.”
Global well-posedness and scattering for the mass critical nonlinear Schr{\"o}dinger equation with mass below the mass of the ground state (2011, unknown): “(See [2], [10], [11], [12], [14], [8], [21], [17], [15], and [16] for more information on the I-method.”
Global well-posedness and scattering for the defocusing, $L^{2}$-critical, nonlinear Schrödinger equation when $d=2$ (2010, unknown): “(See [1], [8], [9], [10], [12], [6], [17], [15], [13], and [14] for more information on the I-method.”
Global well-posedness and scattering for the defocusing, $L^{2}$-critical nonlinear Schrödinger equation when $d =3$ (2009, unknown): “(See [1], [7], [8], [9], [11], [6], [15], [14], [12], and [13] for more information on the I-method.”
The mass-critical nonlinear Schr\"odinger equation with radial data in dimensions three and higher (2007, background): “In dimensions one and two, there has been much work [4, 12, 13, 15, 16, 18, 21, 42] devoted to lowering the regularity of the initial data from H(1) x toward L 2 x(R ) and thus, toward establishing the conjecture.”
Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on ℝ3 (2003, background): “Section 3 revisits the almost conservation law argument in [13], now in the setting of an a priori L4x,t bound on a space-time slab.”
Global well-posedness and scattering for the defocusing, L2-critical, nonlinear Schrödinger equation when d = 1 (2010, background): “See [20, 25, 27] for the almost Morawetz estimates in two dimensions, [26, 29] in one dimension, and [22] for the I-method itself.”