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Global well-posedness of the Majda-Biello system in the resonant case on the real line

Xin Yang

Abstract

We study the Cauchy problem for the following Majda-Biello system in the case $α=4$, where the resonance effect is the most significant, on the real line. \[ \left\{ \begin{array}{rcl} u_{t} + u_{xxx} & = & - v v_x, v_{t} + αv_{xxx} & = & - (uv)_{x}, (u,v)|_{t=0} & = & (u_0,v_0) \in H^{s}(\mathbb{R}) \times H^{s}(\mathbb{R}), \end{array} \right. \quad x \in \mathbb{R}, \, t \in \mathbb{R}. \] For Sobolev regularity $s\in[\frac34, 1)$, we establish global well-posedness by refining the I-method. Previously, the critical index for local well-posedness was known to be $\frac34$, while global well-posedness was only obtained for $s\geq 1$. Our global well-posedness result bridges the gap and matches the threshold in the local theory. The main novelty of our approach is to introduce a pair of distinct $I$-operators, tailored to the resonant structure of the Majda-Biello system with $α=4$. This dual-operator framework allows for pointwise control of the multipliers in the modified energies constructed via the multilinear correction technique. These modified energies are almost conserved and provide effective control over the Sobolev norm of the solution for all time. This new approach has potential applications to other coupled dispersive systems exhibiting strong resonant interactions.

Global well-posedness of the Majda-Biello system in the resonant case on the real line

Abstract

We study the Cauchy problem for the following Majda-Biello system in the case , where the resonance effect is the most significant, on the real line. For Sobolev regularity , we establish global well-posedness by refining the I-method. Previously, the critical index for local well-posedness was known to be , while global well-posedness was only obtained for . Our global well-posedness result bridges the gap and matches the threshold in the local theory. The main novelty of our approach is to introduce a pair of distinct -operators, tailored to the resonant structure of the Majda-Biello system with . This dual-operator framework allows for pointwise control of the multipliers in the modified energies constructed via the multilinear correction technique. These modified energies are almost conserved and provide effective control over the Sobolev norm of the solution for all time. This new approach has potential applications to other coupled dispersive systems exhibiting strong resonant interactions.

Paper Structure

This paper contains 22 sections, 15 theorems, 244 equations.

Table of Contents

  1. Introduction
  2. Literature review and main result
  3. Outline of the approach
  4. Organization of the paper
  5. Modified energies
  6. The first modified energy
  7. Time derivative of energies
  8. The second modified energy
  9. Estimates on the multipliers
  10. Choice of the second $I$-operator $I_2$
  11. Properties of $m_1$
  12. Key estimate on the multiplier $\sigma_3$
  13. Estimate on the multipliers $M_{41}$ and $M_{42}$
  14. Growth estimate of modified energies
  15. Lifespan of local solutions
  16. ...and 7 more sections

Key Result

Theorem 1.1

For the Majda-Biello system (MB) with $\alpha=4$, it is globally well-posed in the space $\mathscr{H}^{s}(\mathbb{R})$ for any $s\in [\frac{3}{4}, 1)$.

Theorems & Definitions (30)

  • Theorem 1.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • ...and 20 more