Well-posedness for a Family of Perturbations of the KDV Equation in Periodic Sobolev Spaces of Negative Order

We establish local well-posedness in Sobolev spaces $H^s(\mathbb{T})$, with $s\geq -1/2$, for the initial value problem issues of the equation $$ u_t + u_{xxx}+\eta Lu + uu_x=0;\; x\in \mathbb{T},\; t\geq0, $$ where $\eta >0$, $(Lu)^{\wedge}(k)=-\Phi(k)\hat{u}(k)$, $k\in \mathbb{Z}$ and $\Phi \in \mathbb{R}$ is bounded above. Particular cases of this problem are the Korteweg-de Vries-Burgers equation for $\Phi(k)=-k^2$, the derivative Korteweg-de Vries-Kuramoto-Sivashinsky equation for $\Phi(k)=k^2-k^4$, and the Ostrovsky-Stepanyams-Tsimring equation for $\Phi(k)=|k|-|k|^3$.