On unconditional well-posedness of modified KdV
Abstract
Bourgain(1993) proved that the periodic modified KdV equation (mKdV) is locally well-posed in Sobolev spave Hs(T), s >= 1/2, by introducing new weighted Sobolev spaces Xs,b, where the uniqueness holds conditionally, namely in the intersection of C([0, T]; Hs) and Xs,b. In this paper, we establish unconditional well-posedness of mKdV in Hs(T), s >= 1/2, i.e. we in addition establish unconditional uniqueness in C([0, T]; Hs), 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.
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