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Well-posedness for KdV-type equations with quadratic nonlinearity

Hiroyuki Hirayama, Shinya Kinoshita, Mamoru Okamoto

Abstract

We consider the Cauchy problem of the KdV-type equation \[ \partial_t u + \frac{1}{3} \partial_x^3 u = c_1 u \partial_x^2u + c_2 (\partial_x u)^2, \quad u(0)=u_0. \] Pilod (2008) showed that the flow map of this Cauchy problem fails to be twice differentiable in the Sobolev space $H^s(\mathbb{R})$ for any $s \in \mathbb{R}$ if $c_1 \neq 0$. By using a gauge transformation, we point out that the contraction mapping theorem is applicable to the Cauchy problem if the initial data are in $H^2(\mathbb{R})$ with bounded primitives. Moreover, we prove that the Cauchy problem is locally well-posed in $H^1(\mathbb{R})$ with bounded primitives.

Well-posedness for KdV-type equations with quadratic nonlinearity

Abstract

We consider the Cauchy problem of the KdV-type equation Pilod (2008) showed that the flow map of this Cauchy problem fails to be twice differentiable in the Sobolev space for any if . By using a gauge transformation, we point out that the contraction mapping theorem is applicable to the Cauchy problem if the initial data are in with bounded primitives. Moreover, we prove that the Cauchy problem is locally well-posed in with bounded primitives.

Paper Structure

This paper contains 13 sections, 9 theorems, 118 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Lemmas
  4. Well-posedness via the contraction mapping theorem
  5. Proof of Theorem \ref{['WP']} with $s=2$
  6. Well-posedness for \ref{['KdV']} in $\mathcal{X}^1$
  7. Proof of Theorem \ref{['WP']} under $c_2=0$
  8. Persistence of regularity
  9. Proof of Theorem \ref{['WP']} without $c_2 = 0$
  10. Well-posedness for the quadratic KdV-type equation
  11. Irregular flow maps
  12. On the condition for initial data
  13. Not $C^2$ in $\mathcal{X}^s$

Key Result

Theorem 1.1

The Cauchy problem for KdV with $u(0) =u_0$ is local-in-time well-posed in $\mathcal{X}^s$ for $s \ge 1$. Moreover, the flow map is (locally) Lipschitz continuous. In addition, the existence time depends only on $\| u_0 \|_{\mathcal{X}^1}$.

Theorems & Definitions (15)

  • Theorem 1.1
  • Remark 1.2
  • Proposition 1.3
  • Theorem 1.4
  • Remark 1.5
  • Lemma 2.1: Lemma 2.4 in KPV89
  • Lemma 2.2
  • Lemma 2.3: Corollary 2.9 in KPV91
  • Lemma 4.1
  • proof
  • ...and 5 more