Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Construction of two-bubble blow-up solutions for the mass-critical gKdV equations

Yang Lan, Xu Yuan

Abstract

For the mass-critical generalized Korteweg-de Vries equation, $$ \partial_{t}u+\partial_{x}\left( \partial_{x}^{2}u+u^{5}\right)=0,\quad (t,x)\in [0,\infty)\times \mathbb{R}.$$ We prove the existence of a global solution that blows up in infinite time and approaches the sum of two decoupled bubbles with opposite signs. The proof is inspired by the techniques developed for the two-dimensional mass-critical NLS equation in a similar context by Martel-Raphaël [37]. The main difficulty originates from the fact that the unstable directions related to scaling are excited by the nonlinear interactions. To overcome this difficulty, a refined approximate solution that involves some non-localized profiles is needed. In particular, a sharp understanding for the interactions between solitons and such profiles is also required.

Construction of two-bubble blow-up solutions for the mass-critical gKdV equations

Abstract

For the mass-critical generalized Korteweg-de Vries equation, We prove the existence of a global solution that blows up in infinite time and approaches the sum of two decoupled bubbles with opposite signs. The proof is inspired by the techniques developed for the two-dimensional mass-critical NLS equation in a similar context by Martel-Raphaël [37]. The main difficulty originates from the fact that the unstable directions related to scaling are excited by the nonlinear interactions. To overcome this difficulty, a refined approximate solution that involves some non-localized profiles is needed. In particular, a sharp understanding for the interactions between solitons and such profiles is also required.
Paper Structure (17 sections, 25 theorems, 586 equations)

This paper contains 17 sections, 25 theorems, 586 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Blow-up description and classification for gKdV equation
  4. Previous results for other models
  5. A Roadmap of the proof
  6. Notation and conventions
  7. Construction of the approximate solution
  8. The linearized operator
  9. Leading order of the nonlinear interaction
  10. Approximate solution
  11. Dynamics close to the two-bubble solution
  12. Modulation of the approximate solution
  13. Backwards uniform estimates
  14. Bootstrap bounds
  15. Energy functional
  16. ...and 2 more sections

Key Result

Theorem 1.1

There exists a global-in-time solution $u\in C\left([0,\infty);H^{1}\right)$ of equ:gKdV that decomposes asymptotically into a sum of two bubbles at infinity: Here, the position parameters $(x_{1},x_{2})$ satisfy In addition, the scaling and position parameters $(\lambda(t),x_{1}(t))$ satisfy In particular, the blow-up rate for $u(t)$ is

Theorems & Definitions (58)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Proposition 2.1: Spectral theory of $\mathcal{L}$
  • proof
  • Remark 2.2
  • Lemma 2.3: Non-localized profiles
  • proof
  • Remark 2.4
  • Remark 2.5
  • ...and 48 more