Bourgain(1993) proved that the periodic modified KdV equation (mKdV) is
locally well-posed in Sobolev spave H^s(T), s >= 1/2, by introducing new
weighted Sobolev spaces X^s,b, where the uniqueness holds conditionally, namely
in the intersection of C([0, T]; H^s) and X^s,b. In this paper, we establish
unconditional well-posedness of mKdV in H^s(T), s >= 1/2, i.e. we in addition
establish unconditional uniqueness in C([0, T]; H^s), s >= 1/2, of solutions to
mKdV. We prove this result via differentiation by parts. For the endpoint case
s = 1/2, we perform careful quinti- and septi-linear estimates after the second
differentiation by parts.Comment: 18 pages, small changes in Section 1. (Remark 1.2 added), to appear
in Int. Math. Res. No
