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Well-posedness for a Family of Perturbations of the KDV Equation in Periodic Sobolev Spaces of Negative Order (1105.2995v3)
Published 16 May 2011 in math.AP
Abstract: We establish local well-posedness in Sobolev spaces $Hs(\mathbb{T})$, 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 $\Phi(k)=-k2$, the derivative Korteweg-de Vries-Kuramoto-Sivashinsky equation for $\Phi(k)=k2-k4$, and the Ostrovsky-Stepanyams-Tsimring equation for $\Phi(k)=|k|-|k|3$.