Korteweg-de Vries equation on T: Difference between revisions

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** [[Solitons]] are asymptotically H^1 stable [MtMe-p3], [<span class="SpellE">MtMe</span>-p].Indeed, the solution decouples into a finite sum of <span class="SpellE">solitons</span> plus dispersive radiation [[references.html#EckShr1988 EckShr1988]]
** [[Solitons]] are asymptotically H^1 stable [MtMe-p3], [<span class="SpellE">MtMe</span>-p].Indeed, the solution decouples into a finite sum of <span class="SpellE">solitons</span> plus dispersive radiation [[references.html#EckShr1988 EckShr1988]]


[[Category:Integrability]]
[[Category:Equations]]  [[Category:Airy]]
[[Category:Equations]]  [[Category:Airy]]

Revision as of 07:37, 31 July 2006

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].