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.