Electron. J. Diff. Eqns., Vol. 2008(2008), No. 02, pp. 1-18.

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

Xavier Carvajal, Mahendra Panthee

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 2, 2008.
Math Subject Classifications: 35A07, 35Q53.
Key Words: Bourgain spaces; KdV equation; local smoothing effect.

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Xavier Carvajal
Instituto de Matemática - UFRJ
Av. Horácio Macedo, Centro de Tecnologia, Cidade Universitária
Ilha do Fundão, Caixa Postal 68530 21941-972 Rio de Janeiro, RJ, Brasil
email: carvajal@im.ufrj.br
Mahendra Panthee
Centro de Análise Matemática, Geometria e Sistemas Dinâmicos
Departamento de Matemática, Instituto Superior Técnico
1049-001 Lisboa, Portugal
email: mpanthee@math.ist.utl.pt

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