Modified Korteweg-de Vries on R: Difference between revisions

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* LWP in <span class="SpellE">H^s</span> for s >= 1/4 [[KnPoVe1993]]
* LWP in <span class="SpellE">H^s</span> for s >= 1/4 [[KnPoVe1993]]
** Was shown for s>3/2 in [[GiTs1989]]
** Was shown for s>3/2 in [[GiTs1989]]
** This is sharp in the <span class="SpellE">focussing</span> case [[KnPoVe-p]], in the sense that the solution map is no longer uniformly continuous for s < 1/4.
** This is sharp in the [[focusing]] case [[KnPoVe-p]], in the sense that the solution map is no longer uniformly continuous for s < 1/4.
*** This has been extended to the <span class="SpellE">defocussing</span> case in [<span class="SpellE">CtCoTa</span>-p], by a high-frequency approximation of <span class="SpellE">mKdV</span> by [schrodinger.html#Cubic NLS on R NLS]. (This high frequency approximation has also been utilized in [[Sch1998]]).
*** This has been extended to the [[defocusing]] case in [[CtCoTa-p]], by a high-frequency approximation of [[mKdV]] by [[Cubic NLS on R|cubic NLS]]. (This high frequency approximation has also been utilized in [[Sch1998]]).
*** Below 1/4 the solution map was known to not be C^3 in [[Bo1993b]], [[Bo1997]].
*** Below 1/4 the solution map was known to not be C^3 in [[Bo1993b]], [[Bo1997]].
** The same result has also been established for the half-line [<span class="SpellE">CoKe</span>-p], assuming boundary data is in H<span class="GramE">^{</span>(s+1)/3} of course.
** The same result has also been established for the half-line [[CoKe-p]], assuming boundary data is in H<span class="GramE">^{</span>(s+1)/3} of course.
** Global weak solutions in L^2 were constructed in [[Ka1983]]. Thus in L^2 one has global existence but no uniform continuity.
** Global weak solutions in L^2 were constructed in [[Ka1983]]. Thus in L^2 one has global existence but no uniform continuity.
** Uniqueness is also known when the initial data lies in the weighted space <x>^{3/8} u_0 in L^2 [[GiTs1989]]
** Uniqueness is also known when the initial data lies in the weighted space <x>^{3/8} u_0 in L^2 [[GiTs1989]]
** LWP has also been demonstrated when <xi>^s hat(u_0) lies in L^{r\u2019} for 4/3 < r <= 2 and s >= ½ - 1/2r [Gr-p4]
** LWP has also been demonstrated when <xi>^s hat(u_0) lies in L^{r\u2019} for 4/3 < r <= 2 and s >= ½ - 1/2r [[Gr-p4]]
* GWP in <span class="SpellE">H^s</span> for s > 1/4 [[CoKeStTkTa-p2]], via the <span class="SpellE">KdV</span> theory and the Miura transform, for both the <span class="SpellE">focussing</span> and <span class="SpellE">defocussing</span> cases.
* GWP in <span class="SpellE">H^s</span> for s > 1/4 [[CoKeStTkTa2003]], via the <span class="SpellE">KdV</span> theory and the Miura transform, for both the <span class="SpellE">focussing</span> and <span class="SpellE">defocussing</span> cases.
** Was proven for s>3/5 in [[FoLiPo1999]]
** Was proven for s>3/5 in [[FoLiPo1999]]
** Is implicit for s >= 1 from [[KnPoVe1993]]
** Is implicit for s >= 1 from [[KnPoVe1993]]
** On the half-line GWP is known when s >= 1 and the boundary data is in H^{11/12}, assuming compatibility and small L^2 norm [<span class="SpellE">CoKe</span>-p]
** On the half-line GWP is known when s >= 1 and the boundary data is in H^{11/12}, assuming compatibility and small L^2 norm [[CoKe-p]]
** GWP for smooth data can also be achieved from inverse scattering methods [<span class="SpellE">BdmFsShp</span>-p]; the same approach also works on an interval [<span class="SpellE">BdmShp</span>-p].
** GWP for smooth data can also be achieved from inverse scattering methods [[BdmFsShp-p]]; the same approach also works on an interval [[BdmShp-p]].
** [[Soliton]]s are asymptotically H^1 stable [MtMe-p3], [<span class="SpellE">MtMe</span>-p]
** [[Soliton]]s are asymptotically H^1 stable [[MtMe-p3]], [[MtMe-p]]


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

Revision as of 22:53, 14 August 2006

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

  • Scaling is s_c = -1/2.
  • LWP in H^s for s >= 1/4 KnPoVe1993
    • Was shown for s>3/2 in GiTs1989
    • This is sharp in the focusing case KnPoVe-p, in the sense that the solution map is no longer uniformly continuous for s < 1/4.
      • This has been extended to the defocusing case in CtCoTa-p, by a high-frequency approximation of mKdV by cubic NLS. (This high frequency approximation has also been utilized in Sch1998).
      • Below 1/4 the solution map was known to not be C^3 in Bo1993b, Bo1997.
    • The same result has also been established for the half-line CoKe-p, assuming boundary data is in H^{(s+1)/3} of course.
    • Global weak solutions in L^2 were constructed in Ka1983. Thus in L^2 one has global existence but no uniform continuity.
    • Uniqueness is also known when the initial data lies in the weighted space <x>^{3/8} u_0 in L^2 GiTs1989
    • LWP has also been demonstrated when <xi>^s hat(u_0) lies in L^{r\u2019} for 4/3 < r <= 2 and s >= ½ - 1/2r Gr-p4
  • GWP in H^s for s > 1/4 CoKeStTkTa2003, via the KdV theory and the Miura transform, for both the focussing and defocussing cases.
    • Was proven for s>3/5 in FoLiPo1999
    • Is implicit for s >= 1 from KnPoVe1993
    • On the half-line GWP is known when s >= 1 and the boundary data is in H^{11/12}, assuming compatibility and small L^2 norm CoKe-p
    • GWP for smooth data can also be achieved from inverse scattering methods BdmFsShp-p; the same approach also works on an interval BdmShp-p.
    • Solitons are asymptotically H^1 stable MtMe-p3, MtMe-p