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A determination of the blowup solutions to the focusing, quintic NLS with mass equal to the mass of the soliton

Benjamin Dodson

Abstract

In this paper we prove that the only blowup solutions to the focusing, quintic nonlinear Schr{ö}dinger equation with mass equal to the mass of the soliton are rescaled solitons or the pseudoconformal transformation of those solitons.

A determination of the blowup solutions to the focusing, quintic NLS with mass equal to the mass of the soliton

Abstract

In this paper we prove that the only blowup solutions to the focusing, quintic nonlinear Schr{ö}dinger equation with mass equal to the mass of the soliton are rescaled solitons or the pseudoconformal transformation of those solitons.

Paper Structure

This paper contains 18 sections, 34 theorems, 504 equations.

Table of Contents

  1. Introduction
  2. Reductions of a symmetric blowup solution
  3. Decomposition of the solution near $Q$
  4. A long time Strichartz estimate
  5. Almost conservation of energy
  6. A frequency localized Morawetz estimate
  7. An $L_{s}^{p}$ bound on $\| \epsilon(s) \|_{L^{2}}$ when $p > 1$
  8. Monotonicity of $\lambda$
  9. Almost monotone $\lambda(t)$
  10. A non-symmetric solution
  11. Reductions of a non-symmetric blowup solution
  12. Decomposition of a non-symmetric solution near $Q$
  13. A long time Strichartz estimate in the nonsymmetric case
  14. Almost conservation of energy for a non-symmetric solution
  15. A frequency localized Morawetz estimate for nonsymmetric $u$
  16. ...and 3 more sections

Key Result

Theorem \oldthetheorem

The initial value problem $(1.1)$ is locally well-posed for any $u_{0} \in L^{2}$.

Theorems & Definitions (75)

  • Theorem \oldthetheorem
  • Remark 1
  • proof
  • Theorem \oldthetheorem
  • proof
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Lemma 1
  • proof
  • ...and 65 more