Well-posedness for KdV-type equations with quadratic nonlinearity

Abstract

We consider the Cauchy problem of the KdV-type equation $$\begin{aligned} \partial _tu + \frac{1}{3} \partial _x^3 u = c_1 u \partial _x^2u + c_2 (\partial _xu)^2, \quad u(0)=u_0. \end{aligned}$$ Pilod (J Differ Equ 245(8):2055–2077, 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 $$c_1 \ne 0$$ . 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.