# Well-posedness for KdV-type equations with quadratic nonlinearity

## TL;DR

This paper establishes local well-posedness for a KdV-type equation with quadratic nonlinearity in certain Sobolev spaces, using gauge transformations to overcome previous regularity limitations and applying contraction mapping techniques.

## Contribution

It demonstrates local well-posedness in H^1 for the KdV-type equation with quadratic nonlinearity, improving understanding of its solution behavior.

## Key findings

- Flow map not twice differentiable in H^s for any s if c_1 ≠ 0.
- Gauge transformation enables application of contraction mapping in H^2 with bounded primitives.
- Proves local well-posedness in H^1 with bounded primitives.

## 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.