Electronic Journal of Differential Equations,
Vol. 2005(2005), No. 118, pp. 1-28.
Title: Stability of energy-critical nonlinear Schrodinger
equations in high dimensions
Authors: Terence Tao (Univ. of California, Los Angeles, CA, USA)
Monica Visan (Univ. of California, Los Angeles, CA, USA)
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 $n > 6$ 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