The well-posedness of 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 $u_t+u_{xxx}+|D_x{|}^{\alpha }u+u^2{u}_x=0$ with $0<\alpha \le 3$. 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 \cite{Tao} for KdV equation.

20 pages

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• Global well-posedness and inviscid limit for the modified Korteweg-de Vries-Burgers equation