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Sharp blow-up stability for self-similar solutions of the modified Korteweg-de Vries equation

Simão Correia, Raphaël Côte

Abstract

We consider the modified Korteweg-de Vries equation. Given a self-similar solution, and a subcritical perturbation of any size, we prove that there exists a unique solution to the equation which behaves at blow-up time as the self-similar solution plus the perturbation. To this end, we develop the first robust analysis in spaces of functions with bounded Fourier transforms. To begin, we prove the local well-posedness in subcritical spaces through an appropriate restriction norm method. As this method is not sufficient to capture the critical self-similar dynamics, we develop an infinite normal form reduction (INFR) to derive time-dependent a priori $L^\infty$ bounds in frequency variables. Both approaches rely on frequency-restricted estimates, which are specific positive multiplier estimates capable of capturing the oscillatory nature of the equation. As a consequence of our analysis, we also prove local well-posedness for small subcritical perturbations of self-similar solutions at positive time.

Sharp blow-up stability for self-similar solutions of the modified Korteweg-de Vries equation

Abstract

We consider the modified Korteweg-de Vries equation. Given a self-similar solution, and a subcritical perturbation of any size, we prove that there exists a unique solution to the equation which behaves at blow-up time as the self-similar solution plus the perturbation. To this end, we develop the first robust analysis in spaces of functions with bounded Fourier transforms. To begin, we prove the local well-posedness in subcritical spaces through an appropriate restriction norm method. As this method is not sufficient to capture the critical self-similar dynamics, we develop an infinite normal form reduction (INFR) to derive time-dependent a priori bounds in frequency variables. Both approaches rely on frequency-restricted estimates, which are specific positive multiplier estimates capable of capturing the oscillatory nature of the equation. As a consequence of our analysis, we also prove local well-posedness for small subcritical perturbations of self-similar solutions at positive time.
Paper Structure (21 sections, 36 theorems, 356 equations, 2 figures)

This paper contains 21 sections, 36 theorems, 356 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Setting and motivation
  3. Main results and description of the proofs
  4. Early steps of the stability proof
  5. Construction of an approximating sequence
  6. A first decomposition of the nonlinearity
  7. Multilinear estimates
  8. Bounds on the self-similar solution
  9. Bounds on F2
  10. Frequency-restricted estimates
  11. Pointwise bounds for the nonlinearity
  12. The infinite normal form reduction
  13. The normal form algorithm
  14. Tree representation of the terms in the INFR
  15. Bounds on the infinite normal form equation
  16. ...and 6 more sections

Key Result

Theorem 1.1

Equation mKdV is locally well-posed inHere, $\widehat{L^\infty}(\langle \xi \rangle^\mu d\xi)=\{u\in \mathcal{S}'(\mathbb{R}^N): \langle \xi \rangle^\mu \hat{u}(\xi)\in L^\infty(\mathbb{R})\}$.$\widehat{L^\infty}(\langle \xi \rangle^\mu d\xi)$, for any $\mu >0$.

Figures (2)

  • Figure 1: The replacement, in the tree on the left, of the $w$ node with an elementary tree yields an admissible tree associated to a badly-behaved term at Step 2.
  • Figure :

Theorems & Definitions (79)

  • Theorem 1.1
  • Proposition 1.2: CCV19
  • Remark 1.1
  • Theorem 1.3: Blow-up stability of self-similar solutions
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.4: Local well-posedness around a self-similar solution at positive times
  • Proposition 2.1
  • proof
  • ...and 69 more