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Bourgain [ 2 ] proved that the periodic modified Korteweg–de Vries (mKdV) equation is locally well-posed in,, by introducing new weighted Sobolev spaces  X   s , b  , where the uniqueness holds  conditionally , namely in. In this paper, we establish  unconditional  well-posedness of mKdV in,, that is, in addition we establish  unconditional  uniqueness in  C ([0, T ]; H   s  ),, of solutions to mKdV. We prove this result via differentiation by parts. For the endpoint case, we perform careful quinti- and septi-linear estimates after the second differentiation by parts.

On Unconditional Well-Posedness of Modified KdV