Korteweg-de Vries equation on T

The local and global well-posedness theory for the Korteweg-de Vries equation on the torus is as follows.

• Scaling is s_c = -3/2.
• C^0 LWP in H^s for s >= -1, assuming u is real [KpTp-p]
  • C^0 LWP in H^s for s >= -5/8 follows (at least in principle) from work on the mKdV equation by [Takaoka and Tsutsumi?]
  • Analytic LWP in H^s for s >= -1/2, in the complex case references.html#KnPoVe1996 KnPoVe1996. In addition to the usual bilinear estimate, one needs a linear estimate to keep the solution in H^s for t>0.
  • Analytic LWP was proven for s >= 0 in references.html#Bo1993b Bo1993b.
  • Analytic ill posedness at s<-1/2, even in the real case references.html#Bo1997 Bo1997
    • This has been refined to failure of uniform continuity at s<-1/2 [CtCoTa-p]
  • Remark: s=-1/2 is the symplectic regularity, and so the machinery of infinite-dimensional symplectic geometry applies once one has a continuous flow, although there are some technicalities involving approximating KdV by a suitable symplectic finite-dimensional flow. In particular one has symplectic non-squeezing [CoKeStTkTa-p9], references.html#Bo1999 Bo1999.
• C^0 GWP in H^s for s >= -1, in the real case [KpTp-p].
  • Analytic GWP in H^s in the real case for s >= -1/2 references.html#CoKeStTaTk-p2 CoKeStTkTa-p2; see also references.html#CoKeStTaTk-p3 CoKeStTkTa-p3.
  • A short proof for the s > -3/10 case is in references.html#CoKeStTaTk-p2a CoKeStTkTa-p2a
  • Was proven for s >= 0 in references.html#Bo1993b Bo1993b.
  • GWP for real initial data which are measures of small norm references.html#Bo1997 Bo1997 The small norm restriction is presumably technical.
    • Remark: measures have the same scaling as H^{-1/2}, but neither space includes the other. (Measures are in H^{-1/2-\eps} though).
  • One has GWP for real random data whose Fourier coefficients decay like 1/|k| (times a Gaussian random variable) references.html#Bo1995c Bo1995c. Indeed one has an invariant measure.
  • Solitons are asymptotically H^1 stable [MtMe-p3], [MtMe-p].Indeed, the solution decouples into a finite sum of solitons plus dispersive radiation references.html#EckShr1988 EckShr1988