Korteweg-de Vries equation on T

From DispersiveWiki

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 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 Bo1993b.
    • Analytic ill posedness at s<-1/2, even in the real case 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, 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 CoKeStTkTa2003; see also CoKeStTkTa-p3.
    • A short proof for the s > -3/10 case is in CoKeStTkTa2001
    • Was proven for s >= 0 in Bo1993b.
    • GWP for real initial data which are measures of small norm 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) 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 EckShr1983.