On the well-posedness of the Cauchy problem for dissipative modified Korteweg-de Vries equations

Abstract

In this paper we consider some dissipative versions of the modified Korteweg--de~Vries equation $u_t+u_{xxx}+|D_x|^{\alpha}u+u^2u_x=0$ with $0 <\alpha\leq 3$. We prove some well-posedness results on the associated Cauchy problem in the Sobolev spaces $H^s({ \mathbb R})$ for $s>1/4-\alpha/4$ on the basis of the $[k;\,Z]-$multiplier norm estimate obtained by Tao in [11] for KdV equation.