On unconditional well-posedness of modified KdV

By Soonsik Kwon, Tadahiro Oh. 2010

67 citations recorded by the source corpus.

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.

Subfields: Control and Systems Engineering, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/c1ba7996cd35815cc8d62cc0f3c928a071210f22

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In Control and Systems Engineering

In Mathematical Physics

In Statistical and Nonlinear Physics

Citations in the corpus

An optimal regularity result on the quasi-invariant Gaussian measures for the cubic fourth order nonlinear Schr\"odinger equation (2017, unknown): “Our normal form approach is analogous to the approach employed in [1, 28, 18].”
Remarks on the Gibbs measures for nonlinear dispersive equations (2014, background): “See Christ [18], Colliander-Oh [20], Kwon-Oh [30], and Bourgain [5] in the 2-dimensional case.”