The quintic NLS on perturbations of $\mathbf{R}^3$

Casey Jao

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arXiv · 1607.03851

The quintic NLS on perturbations of $\mathbf{R}^3$

Casey Jao

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Abstract

Consider the defocusing quintic nonlinear Schr\"{o}dinger equation on $\mathbf{R}^3$ with initial data in the energy space. This problem is "energy-critical" in view of a certain scale-invariance, which is a main source of difficulty in the analysis of this equation. It is a nontrivial fact that all finite-energy solutions scatter to linear solutions. We show that this remains true under small compact deformations of the Euclidean metric. Our main new ingredient is a long-time microlocal weak dispersive estimate that accounts for the refocusing of geodesics.

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Casey Jao. 2016-07-13. The quintic NLS on perturbations of $\mathbf{R}^3$. https://arxiv.org/abs/1607.03851

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