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Global well-posedness for the Cauchy problem of the Zakharov-Kuznetsov equation on cylindrical spaces

Satoshi Osawa, Hideo Takaoka

TL;DR

This work establishes global well-posedness for the Zakharov–Kuznetsov equation on the cylinder $\mathbb{R}\times \mathbb{T}$ at low regularity, proving GWP in $H^s(\mathbb{R}\times \mathbb{T})$ for $s>\frac{29}{31}$ using the I-method. The authors construct a rescaled problem on $\mathbb{R}\times \mathbb{T}_{\lambda}$, introduce the smoothing operator $I$ with multiplier $m(\zeta)$ to elevate rough data to the $H^1$-level, and derive sharp bilinear estimates in the Bourgain spaces $X^{s,b}_{\lambda}$ to close a contraction. They define a modified energy $E^{\lambda}[Iu^{\lambda}]$ and show its increment is almost conserved, with quantitative bounds that depend on $N$ and the regularity threshold. By balancing the energy increment with a rescaling argument, they extend local solutions to global ones and obtain polynomial-in-time growth for the $H^s$-norm, contributing to global well-posedness theory for dispersive equations on cylindrical domains.

Abstract

We prove that the Zakharov-Kuznetsov equation on cylindrical spaces is globally well-posed below the energy norm. As is known, local well-posedness below energy space was obtained by the first author. We adapt I-method to extend the solutions globally in time. Using modified energies, we obtain the polynomial bounds on the $H^s$ growth for the global solutions.

Global well-posedness for the Cauchy problem of the Zakharov-Kuznetsov equation on cylindrical spaces

TL;DR

This work establishes global well-posedness for the Zakharov–Kuznetsov equation on the cylinder at low regularity, proving GWP in for using the I-method. The authors construct a rescaled problem on , introduce the smoothing operator with multiplier to elevate rough data to the -level, and derive sharp bilinear estimates in the Bourgain spaces to close a contraction. They define a modified energy and show its increment is almost conserved, with quantitative bounds that depend on and the regularity threshold. By balancing the energy increment with a rescaling argument, they extend local solutions to global ones and obtain polynomial-in-time growth for the -norm, contributing to global well-posedness theory for dispersive equations on cylindrical domains.

Abstract

We prove that the Zakharov-Kuznetsov equation on cylindrical spaces is globally well-posed below the energy norm. As is known, local well-posedness below energy space was obtained by the first author. We adapt I-method to extend the solutions globally in time. Using modified energies, we obtain the polynomial bounds on the growth for the global solutions.
Paper Structure (5 sections, 14 theorems, 180 equations)

This paper contains 5 sections, 14 theorems, 180 equations.

Table of Contents

  1. Introduction
  2. Some notations of function spaces
  3. Rescaled solutions
  4. Modified energy
  5. Proof of main theorem

Key Result

Theorem 1.1

The initial value problem of ZK is globally well-posed in $H^s(\mathbb{R} \times \mathbb{T})$ for $s>29/31$. In other words, for any $u_0 \in H^s(\mathbb{R} \times \mathbb{T})$, for all $T>0$, there exists a unique solution $u$ of ZK such that Moreover, for all $0<T' < T$, there exists a neighborhood $\mathcal{U}$ of $u_0$ in $H^s(\mathbb{R} \times \mathbb{T})$ such that the data-to-solution map

Theorems & Definitions (26)

  • Theorem 1.1
  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Proposition 3.1: A variant of local well-posedness
  • Lemma 3.2: Lemma 3.7 in Molinet
  • Lemma 3.3: Lemma 3.8 in Molinet
  • Lemma 3.4
  • ...and 16 more