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

Abstract

We establish local well-posedness in Sobolev spaces $H^s(\mathbb{T})$, with $s\ge -1/2$, for the initial value problem issues of the equation $$u_t+u_{xxx}+\eta Lu+u{u}_x=0;x\in \mathbb{T},t\ge 0,$$ 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 $\Phi (k)=|k|-|k{|}^3$.

arXiv comment21 pages. arXiv admin note: text overlap with arXiv:math/0110045 by other authors