Korteweg-de Vries equation on T: Difference between revisions
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*** ''Remark''<nowiki>: measures have the same scaling as H</nowiki><span class="GramE">^{</span>-1/2}, but neither space includes the other. (Measures are in H<span class="GramE">^{</span>-1/2-\eps} though). | *** ''Remark''<nowiki>: measures have the same scaling as H</nowiki><span class="GramE">^{</span>-1/2}, but neither space includes the other. (Measures are in H<span class="GramE">^{</span>-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. | ** 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], [<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:Equations]] [[Category:Airy]] | [[Category:Equations]] [[Category:Airy]] | ||
Revision as of 01:12, 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].
- 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
