Electronic Journal of Differential Equations,
Vol. 2008(2008), No. 02, pp. 1-18.

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

Authors: Xavier Carvajal (UFRJ, Rio de Janeiro, Brasil)
         Mahendra Panthee (Inst. Superior Tecnico, Lisboa, Portugal)

Abstract: 
 We  study  some well-posedness issues of  the initial value problem
 associated with the equation
 $$
 u_t+u_{xxx}+\eta Lu+uu_x=0, \quad x \in \mathbb{R}, \; t\geq 0,
 $$
 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 $\Phi(\xi)=-\xi^2$.
 
Submitted August 1, 2007. Published January 02, 2008.
Math Subject Classifications: 35A07, 35Q53.
Key Words: Bourgain spaces; KdV equation; local smoothing effect.