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 $\Phi(k)=|k|-|k|^3$.
Categories: math.AP