Geodesic
Well-posedness for some perturbations of the KdV equation with low regularity data
Electronic journal of differential equations, Tome 2008 (2008)
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 $\Phi(\xi)=-\xi^2$.
Classification : 35A07, 35Q53
Keywords: Bourgain spaces, KdV equation, local smoothing effect 
@article{EJDE_2008__2008__a18,
     author = {Carvajal,  Xavier and Panthee,  Mahendra},
     title = {Well-posedness for some perturbations of the {KdV} equation with low regularity data},
     journal = {Electronic journal of differential equations},
     year = {2008},
     volume = {2008},
     zbl = {1136.35076},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2008__2008__a18/}
}
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%A Panthee,  Mahendra
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Carvajal,  Xavier; Panthee,  Mahendra. Well-posedness for some perturbations of the KdV equation with low regularity data. Electronic journal of differential equations, Tome 2008 (2008). http://geodesic.mathdoc.fr/item/EJDE_2008__2008__a18/