On unconditional well-posedness of modified KdV

Details On unconditional well-posedness of modified KdV On unconditional well-posedness of modified KdV

2010-07-01

arxiv.org/abs/1007.0270v2

Mathematics

Analysis of PDEs

18 pages, small changes in Section 1. (Remark 1.2 added), to appear in Int. Math. Res. Not

Scientific paper

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.

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