Stability of energy-critical nonlinear Schrodinger equations in high dimensions

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.

Keywords

• Local well-posedness
• uniform well-posedness
• scattering theory
• Strichartz estimates.