On Unconditional Well-Posedness of Modified KdV

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Bourgain [2] proved that the periodic modified Korteweg-de Vries (mKdV) equation is locally well-posed in H-s(T), s >= 1/2, by introducing new weighted Sobolev spaces X-s,X-b, where the uniqueness holds conditionally, namely in C([0, T]; H-s) boolean AND X-s,X-1/2 ([0, T] x T). In this paper, we establish unconditional well-posedness of mKdV in H-s(T), s >= 1/2, that is, in addition we 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.
Publisher
OXFORD UNIV PRESS
Issue Date
2012
Language
English
Article Type
Article
Keywords

H-S; EQUATION; SCHRODINGER; UNIQUENESS

Citation

INTERNATIONAL MATHEMATICS RESEARCH NOTICES, no.15, pp.3509 - 3534

ISSN
1073-7928
DOI
10.1093/imrn/rnr156
URI
http://hdl.handle.net/10203/103987
Appears in Collection
MA-Journal Papers(저널논문)
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