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A universal bound on the blow-up rate for the focusing mass-critical nonlinear Schrödinger equation

Beomjong Kwak, Soonsik Kwon

Abstract

In this paper, we investigate a universal blow-up bound for the focusing mass-critical nonlinear Schrödinger equation for general initial data in $L^2(\mathbb R^d)$, extending previous knowledge for mass near the ground-state threshold due to Merle and Raphaël. The main results are twofold. First, we show the nonexistence of self-similar rate blow-up solutions. Second, under radial symmetry, we establish the sharp log--log correction to the self-similar bound on the blow-up rate. The proofs are based on a new analysis of general blow-up solutions, which does not rely on any ansatz or variational structure.

A universal bound on the blow-up rate for the focusing mass-critical nonlinear Schrödinger equation

Abstract

In this paper, we investigate a universal blow-up bound for the focusing mass-critical nonlinear Schrödinger equation for general initial data in , extending previous knowledge for mass near the ground-state threshold due to Merle and Raphaël. The main results are twofold. First, we show the nonexistence of self-similar rate blow-up solutions. Second, under radial symmetry, we establish the sharp log--log correction to the self-similar bound on the blow-up rate. The proofs are based on a new analysis of general blow-up solutions, which does not rely on any ansatz or variational structure.
Paper Structure (13 sections, 26 theorems, 267 equations)

This paper contains 13 sections, 26 theorems, 267 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. The Schrödinger operator and the virial identity
  5. Local smoothing estimates
  6. In/out decomposition and radial Strichartz estimates
  7. Carleman inequalities
  8. Nonexistence of self-similar blow-up rate solutions
  9. Reduction to almost periodic solutions modulo self-similar symmetry
  10. Nonexistence of almost periodic solution modulo self-similar symmetry
  11. Decomposition of a radial solution to (NLS)
  12. Quantification of the mass radiation
  13. Proofs of Theorem \ref{['thm:loglog']}, Corollary \ref{['cor:loglog la Bourgain']}, and Theorem \ref{['thm:saturate loglog']}

Key Result

Theorem 1.1

Let $d\ge 1$. Let $u(t)\in L^2$ be a solution to eq:NLS that blows up at time $T<\infty$. For any $\epsilon>0$, we have

Theorems & Definitions (55)

  • Theorem 1.1: Nonexistence of self-similar blow-up
  • Theorem 1.2: Log--log blow-up bound for radial solutions
  • Remark 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Remark 2.3
  • ...and 45 more