Abstract. In this paper we consider some dissipative versions of the modified Korteweg de Vries equation ut+uxxx+ |Dx| u+uux = 0 with 0 < α ≤ 3. We prove some well-posedness results on the associated Cauchy problem in the Sobolev spaces Hs(R) for s > 1/4−α/4 on the basis of the [k; Z]−multiplier norm estimate obtained by Tao in [9] for KdV equation. 2000… (More)