Global well-posedness for dissipative Korteweg-de Vries equations

Abstract

This paper is devoted to the well-posedness for dissipative KdV equations $u_t+u_{xxx}+|D_x{|}^{2\alpha }u+u{u}_x=0$, $0<\alpha \le 1$. An optimal bilinear estimate is obtained in Bourgain's type spaces, which provides global well-posedness in $H^s(\mathbb{R})$, $s>-3/4$ for $\alpha \le 1/2$ and $s>-3/(5-2\alpha )$ for $\alpha >1/2$.

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