Well-posedness for some perturbations of the KdV equation with low regularity data

We study some well-posedness issues of the initial value problem associated with the equation $$ u_t+u_{xxx}+eta Lu+uu_x=0, quad x in mathbb{R}, ; tgeq 0, $$ where $eta>0$, $widehat{Lu}(xi)=-Phi(xi)hat{u}(xi)$ and $Phi in mathbb{R}$ is bounded above. Using the theory developed by Bourgain and Kenig, Ponce and Vega, we prove that the initial value problem is locally well-posed for given data in Sobolev spaces $H^s(mathbb{R})$ with regularity below $L^2$. Examples of this model are the Ostrovsky-Stepanyams-Tsimring equation for $Phi(xi)=|xi|-|xi|^3$, the derivative Korteweg-de Vries-Kuramoto-Sivashinsky equation for $Phi(xi)=xi^2-xi^4$, and the Korteweg-de Vries-Burguers equation for $Phi(xi)=-xi^2$