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A note on the well-posedness of the quartic Zakharov-Kuznetsov equation on $\mathbb{R} \times \mathbb{T}$

Jakob Nowicki-Koth

Abstract

By using a bilinear smoothing estimate recently developed in [12], together with several linear Strichartz-type estimates established therein, we improve the threshold for local well-posedness of the quartic Zakharov-Kuznetsov equation and prove that it is locally well-posed in $H^s(\mathbb{R} \times \mathbb{T})$ for all $s > \frac{1}{2}$.

A note on the well-posedness of the quartic Zakharov-Kuznetsov equation on $\mathbb{R} \times \mathbb{T}$

Abstract

By using a bilinear smoothing estimate recently developed in [12], together with several linear Strichartz-type estimates established therein, we improve the threshold for local well-posedness of the quartic Zakharov-Kuznetsov equation and prove that it is locally well-posed in for all .
Paper Structure (3 sections, 2 theorems, 47 equations)

This paper contains 3 sections, 2 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Notation, function spaces, and Strichartz-type estimates
  3. Proof of Theorem \ref{['LWP3gZK']}

Key Result

Theorem 1.1

The Cauchy problem $\mathrm{CP}$ for the quartic Zakharov-Kuznetsov equation is locally well-posed for every $s > \frac{1}{2}$. That is, for each $s > \frac{1}{2}$ and every $u_0 \in H^s({\mathbb{R}\times\mathbb{T}})$, there exist a lifespan $\delta = \delta(\norm{u_0}_{H^{\frac{1}{2}+}}) > 0$ and a to $\mathrm{CP}$. Moreover, for every $\tilde{\delta} \in \intoo{0,\delta}$, there exists a neighbo

Theorems & Definitions (3)

  • Theorem 1.1
  • Proposition 3.1
  • proof