Electron. J. Diff. Eqns., Vol. 2005(2005), No. 118, pp. 1-28.

Stability of energy-critical nonlinear Schrodinger equations in high dimensions

Terence Tao, Monica Visan

Abstract:
We develop the existence, uniqueness, continuity, stability, and scattering theory for energy-critical nonlinear Schrodinger equations in dimensions $n \geq 3$, for solutions which have large, but finite, energy and large, but finite, Strichartz norms. For dimensions $n \leq 6$, this theory is a standard extension of the small data well-posedness theory based on iteration in Strichartz spaces. However, in dimensions n greater than 6 there is an obstruction to this approach because of the subquadratic nature of the nonlinearity (which makes the derivative of the nonlinearity non-Lipschitz). We resolve this by iterating in exotic Strichartz spaces instead. The theory developed here will be applied in a subsequent paper of the second author, [21], to establish global well-posedness and scattering for the defocusing energy-critical equation for large energy data.

Submitted July 2, 2005. Published October 26, 2005.
Math Subject Classifications: 35J10.
Key Words: Local well-posedness; uniform well-posedness; scattering theory; Strichartz estimates

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Terence Tao
Department of Mathematics
University of California
Los Angeles, CA 90095-1555, USA
email: tao@math.ucla.edu
Monica Visan
Department of Mathematics
University of California
Los Angeles, CA 90095-1555, USA
email: mvisan@math.ucla.edu

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