Well-posedness for KdV-type equations with quadratic nonlinearity

We consider the Cauchy problem of the KdV-type equation partial differential tu+13 partial differential x3u=c1u partial differential x2u+c2( partial differential xu)2,u(0)=u0.$$\begin{aligned} \partial _tu + \frac{1}{3} \partial _x<^>3 u = c_1 u \partial _x<^>2u + c_2 (\partial _xu)<^>2, \quad u(0)=u_0. \end{aligned}$$\end{document}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 spaceHs(R)for anys is an element of Rifc1 not equal 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 inH2(R)with bounded primitives. Moreover, we prove that the Cauchy problem is locally well-posed inH1(R) with bounded primitives.