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Local and global well-posedness for the 2D Zakharov-Kuznetsov-Burgers equation in low regularity Sobolev space

Hiroyuki Hirayama

Abstract

In the present paper, we consider the Cauchy problem of the 2D Zakharov-Kuznetsov-Burgers (ZKB) equation, which has the dissipative term $-\partial_x^2u$. This is known that the 2D Zakharov-Kuznetsov equation is well-posed in $H^s(\mathbb{R}^2)$ for $s>1/2$, and the 2D nonlinear parabolic equation with quadratic derivative nonlinearity is well-posed in $H^s(\mathbb{R}^2)$ for $s\ge 0$. By using the Fourier restriction norm with dissipative effect, we prove the well-posedness for ZKB equation in $H^s(\mathbb{R}^2)$ for $s>-1/2$.

Local and global well-posedness for the 2D Zakharov-Kuznetsov-Burgers equation in low regularity Sobolev space

Abstract

In the present paper, we consider the Cauchy problem of the 2D Zakharov-Kuznetsov-Burgers (ZKB) equation, which has the dissipative term . This is known that the 2D Zakharov-Kuznetsov equation is well-posed in for , and the 2D nonlinear parabolic equation with quadratic derivative nonlinearity is well-posed in for . By using the Fourier restriction norm with dissipative effect, we prove the well-posedness for ZKB equation in for .

Paper Structure

This paper contains 4 sections, 15 theorems, 155 equations.

Table of Contents

  1. Introduction
  2. Function space and linear estimate
  3. Bilinear estimate
  4. Proof of the well-posedness

Key Result

Theorem 1.1

Let $s>-\frac{1}{2}$. Then (ZKB_sym) is locally well-posed in $H^s({\mathbb R}^2)$. (Therefore (ZKB) is also locally well-posed in $H^s({\mathbb R}^2)$.) More precisely, for any $v_0\in H^s({\mathbb R}^2)$, there exist $T>0$, and an unique solution $v\in X^{s,\frac{1}{2},1}_T\ (\hookrightarrow C([0,

Theorems & Definitions (31)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Definition 2.1
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Proposition 3.1
  • ...and 21 more