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Low-regularity global well-posedness theory for the generalized Zakharov-Kuznetsov equation on $\mathbb{R} \times \mathbb{T}$ and polynomial growth of higher Sobolev norms

Jakob Nowicki-Koth

Abstract

We address the Cauchy problem for the $k$-generalized Zakharov-Kuznetsov equation ($k$-gZK) posed on $\mathbb{R}^2$ and on $\mathbb{R} \times \mathbb{T}$. By applying established and recently developed linear and bilinear Strichartz-type estimates within the framework of the $I$-method, we obtain the following results: $\bullet$ The Zakharov-Kuznetsov equation is globally well-posed in $H^s(\mathbb{R} \times \mathbb{T})$ for every $s>\frac{11}{13}$. $\bullet$ The modified Zakharov-Kuznetsov equation is globally well-posed in $H^s(\mathbb{R}^2)$ for every $s>\frac{2}{3}$ and in $H^s(\mathbb{R} \times \mathbb{T})$ for every $s>\frac{36}{49}$. Moreover, we show that the $H^s(\mathbb{R} \times \mathbb{T})$-norm of smooth global real-valued solutions of $k$-gZK grows at most polynomially in time for every $k\geq 1$.

Low-regularity global well-posedness theory for the generalized Zakharov-Kuznetsov equation on $\mathbb{R} \times \mathbb{T}$ and polynomial growth of higher Sobolev norms

Abstract

We address the Cauchy problem for the -generalized Zakharov-Kuznetsov equation (-gZK) posed on and on . By applying established and recently developed linear and bilinear Strichartz-type estimates within the framework of the -method, we obtain the following results: The Zakharov-Kuznetsov equation is globally well-posed in for every . The modified Zakharov-Kuznetsov equation is globally well-posed in for every and in for every . Moreover, we show that the -norm of smooth global real-valued solutions of -gZK grows at most polynomially in time for every .

Paper Structure

This paper contains 13 sections, 26 theorems, 392 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Function spaces
  5. Rescaled Strichartz-type estimates
  6. global well-posedness below $H^1$
  7. The $I$-method: setup and general arguments
  8. Improved GWP for ZK on $\mathbb{R} \times \mathbb{T}$
  9. (Improved) GWP for mZK on $\mathbb{R}^2$ and $\mathbb{R} \times \mathbb{T}$
  10. Polynomial growth bounds for generalized ZK on $\mathbb{R} \times \mathbb{T}$
  11. General arguments
  12. Key estimate in the case $k=1$
  13. Key estimate in the case $k\geq2$

Key Result

Theorem 1.1

The Cauchy problem $\mathrm{(CP}_{1, \mathbb{R} \times \mathbb{T}} \mathrm{)}$ is globally well-posed for every $s > \frac{11}{13}$. That is, for every $s > \frac{11}{13}$, every $u_0 \in H^s(\mathbb{R} \times \mathbb{T})$, and every prescribed lifespan $T > 0$, there exists a unique solution of $\mathrm{(CP}_{1, \mathbb{R} \times \mathbb{T}} \mathrm{)}$. This solution belongs to $C(\intcc{-T,T},

Theorems & Definitions (65)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 3.1
  • proof
  • Remark 3.2
  • Proposition 3.3
  • proof
  • Remark 3.4
  • Proposition 3.5
  • ...and 55 more