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

Casey Jao

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

Analysis of PDEs 2016-08-26 v2

Authors: Casey Jao

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.

Keywords

quintic nonlinear Schrödinger equation energy-critical scattering microlocal weak dispersive estimate Euclidean metric perturbation geodesic refocusing

Cite

@article{arxiv.1607.03851,
  title  = {The quintic NLS on perturbations of $\mathbf{R}^3$},
  author = {Casey Jao},
  journal= {arXiv preprint arXiv:1607.03851},
  year   = {2016}
}

Comments

Added an appendix detailing the L^1->L^\infty estimate previously mentioned in passing; minor expository improvements

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