Difference between revisions of "Korteweg-de Vries equation on T"

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* C^0 LWP in <span class="SpellE">H^s</span> for s >= -1, assuming u is real [<span class="SpellE">KpTp</span>-p]
 
* C^0 LWP in <span class="SpellE">H^s</span> for s >= -1, assuming u is real [<span class="SpellE">KpTp</span>-p]
 
** C^0 LWP in <span class="SpellE">H^s</span> for s >= -5/8 follows (at least in principle) from work on the <span class="SpellE">mKdV</span> equation by [Takaoka and <span class="SpellE">Tsutsumi</span>?]
 
** C^0 LWP in <span class="SpellE">H^s</span> for s >= -5/8 follows (at least in principle) from work on the <span class="SpellE">mKdV</span> equation by [Takaoka and <span class="SpellE">Tsutsumi</span>?]
** Analytic LWP in <span class="SpellE">H^s</span> 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 <span class="SpellE">H^s</span> for t>0.
+
** Analytic LWP in <span class="SpellE">H^s</span> for s >= -1/2, in the complex case [[Bibliography#KnPoVe1996|KnPoVe1996]]. In addition to the usual bilinear estimate, one needs a linear estimate to keep the solution in <span class="SpellE">H^s</span> for t>0.
** Analytic LWP was proven for s >= 0 in [[references.html#Bo1993b Bo1993b]].
+
** Analytic LWP was proven for s >= 0 in [[Bibliography#Bo1993b|Bo1993b]].
** Analytic ill <span class="SpellE">posedness</span> at s<-1/2, even in the real case [[references.html#Bo1997 Bo1997]]
+
** Analytic ill <span class="SpellE">posedness</span> at s<-1/2, even in the real case [[Bibliography#Bo1997|Bo1997]]
 
*** This has been refined to failure of uniform continuity at s<-1/2 [<span class="SpellE">CtCoTa</span>-p]
 
*** This has been refined to failure of uniform continuity at s<-1/2 [<span class="SpellE">CtCoTa</span>-p]
** Remark: s=-1/2 is the <span class="SpellE">symplectic</span> regularity, and so the machinery of infinite-dimensional <span class="SpellE">symplectic</span> geometry applies once one has a continuous flow, although there are some technicalities involving approximating <span class="SpellE">KdV</span> by a suitable <span class="SpellE">symplectic</span> finite-dimensional flow. In particular one has <span class="SpellE">symplectic</span> non-squeezing [CoKeStTkTa-p9], [[references.html#Bo1999 Bo1999]].
+
** Remark: s=-1/2 is the <span class="SpellE">symplectic</span> regularity, and so the machinery of infinite-dimensional <span class="SpellE">symplectic</span> geometry applies once one has a continuous flow, although there are some technicalities involving approximating <span class="SpellE">KdV</span> by a suitable <span class="SpellE">symplectic</span> finite-dimensional flow. In particular one has <span class="SpellE">symplectic</span> non-squeezing [CoKeStTkTa-p9], [[Bibliography#Bo1999|Bo1999]].
 
* C^0 GWP in <span class="SpellE">H^s</span> for s >= -1, in the real case [<span class="SpellE">KpTp</span>-p].
 
* C^0 GWP in <span class="SpellE">H^s</span> for s >= -1, in the real case [<span class="SpellE">KpTp</span>-p].
** Analytic GWP in <span class="SpellE">H^s</span> in the real case for s >= -1/2 [[references.html#CoKeStTaTk-p2 CoKeStTkTa-p2]]; see also [[references.html#CoKeStTaTk-p3 CoKeStTkTa-p3]].
+
** Analytic GWP in <span class="SpellE">H^s</span> in the real case for s >= -1/2 [[Bibliography#CoKeStTaTk-p2 |CoKeStTkTa-p2]]; see also [[Bibliography#CoKeStTaTk-p3 |CoKeStTkTa-p3]].
** A short proof for the s > -3/10 case is in [[references.html#CoKeStTaTk-p2a CoKeStTkTa-p2a]]
+
** A short proof for the s > -3/10 case is in [[Bibliography#CoKeStTaTk-p2a |CoKeStTkTa-p2a]]
** Was proven for s >= 0 in [[references.html#Bo1993b Bo1993b]].
+
** Was proven for s >= 0 in [[Bibliography#Bo1993b|Bo1993b]].
** GWP for real initial data which are measures of small norm [[references.html#Bo1997 Bo1997]] <span class="GramE">The</span> small norm restriction is presumably technical.
+
** GWP for real initial data which are measures of small norm [[Bibliography#Bo1997|Bo1997]] <span class="GramE">The</span> small norm restriction is presumably technical.
 
*** ''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) [[Bibliography#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]]
+
** [[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 [[Bibliography#EckShr1988|EckShr1988]]
  
 
[[Category:Integrability]]
 
[[Category:Integrability]]
 
[[Category:Equations]]  [[Category:Airy]]
 
[[Category:Equations]]  [[Category:Airy]]

Revision as of 16:23, 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 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 CoKeStTkTa-p2; see also CoKeStTkTa-p3.
    • A short proof for the s > -3/10 case is in CoKeStTkTa-p2a
    • 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 EckShr1988