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Blow-up criteria below scaling for defocusing energy-supercritical NLS and quantitative global scattering bounds

Aynur Bulut

Abstract

We establish quantitative blow-up criteria below the scaling threshold for radially symmetric solutions to the defocusing nonlinear Schrödinger equation with nonlinearity $|u|^6u$. This provides to our knowledge the first generic results distinguishing potential blow-up solutions of the defocusing equation from many of the known examples of blow-up in the focusing case. Our main tool is a quantitative version of a result showing that uniform bounds on $L^2$-based critical Sobolev norms imply scattering estimates. As another application of our techniques, we establish a variant which allows for slow growth in the critical norm. We show that if the critical Sobolev norm on compact time intervals is controlled by a slowly growing quantity depending on the Stricharz norm, then the solution can be extended globally in time, with a corresponding scattering estimate.

Blow-up criteria below scaling for defocusing energy-supercritical NLS and quantitative global scattering bounds

Abstract

We establish quantitative blow-up criteria below the scaling threshold for radially symmetric solutions to the defocusing nonlinear Schrödinger equation with nonlinearity . This provides to our knowledge the first generic results distinguishing potential blow-up solutions of the defocusing equation from many of the known examples of blow-up in the focusing case. Our main tool is a quantitative version of a result showing that uniform bounds on -based critical Sobolev norms imply scattering estimates. As another application of our techniques, we establish a variant which allows for slow growth in the critical norm. We show that if the critical Sobolev norm on compact time intervals is controlled by a slowly growing quantity depending on the Stricharz norm, then the solution can be extended globally in time, with a corresponding scattering estimate.

Paper Structure

This paper contains 11 sections, 14 theorems, 125 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and Strichartz estimates
  3. Proof of Proposition \ref{['prop-uniform']}
  4. Preliminary construction
  5. Classification of intervals
  6. Concentration bound
  7. Morawetz control
  8. Recursive control over unexceptional intervals
  9. Conclusion of the argument
  10. Proof of Theorem \ref{['thm1']}
  11. Proof of Corollary $\ref{['cor1']}$: allowing for growth

Key Result

Theorem \oldthetheorem

There exists $C>0$ such that for each $E\geq 1$ and $M>0$ there exists $\delta_0=\delta_0(E,M)>0$ with the following property: For all radially symmetric initial data $u_0\in \dot{H}_x^{s_c}(\mathbb{R}^3)\cap \dot{H}_x^{s_c+1}(\mathbb{R}^3)$, with if $0<\delta<\delta_0$ and is a solution with maximal-lifespan $I_{\textrm{max}}$ to (eq1) which satisfies, then $I_{\textrm{max}}=\mathbb{R}$ and

Theorems & Definitions (26)

  • Theorem \oldthetheorem
  • Proposition \oldthetheorem
  • Corollary \oldthetheorem
  • Proposition \oldthetheorem: Spatially localized Morawetz estimate
  • proof
  • Remark \oldthetheorem: Absorbing $E$ into expressions involving $\eta$
  • Lemma \oldthetheorem
  • proof
  • Corollary \oldthetheorem
  • Lemma \oldthetheorem
  • ...and 16 more