Difference between revisions of "Modified Korteweg-de Vries on R"

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The local and global [[well-posedness]] theory for the [[modified Korteweg-de Vries equation]] on the line and half-line is as follows.
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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 <span class="SpellE">s_c</span> = -1/2.
 
* Scaling is <span class="SpellE">s_c</span> = -1/2.
* LWP in <span class="SpellE">H^s</span> for s >= 1/4 [[references.html#KnPoVe1993 KnPoVe1993]]
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* LWP in <span class="SpellE">H^s</span> for s >= 1/4 [[Bibliography#KnPoVe1993|KnPoVe1993]]
** Was shown for s>3/2 in [[references.html#GiTs1989 GiTs1989]]
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** Was shown for s>3/2 in [[Bibliography#GiTs1989|GiTs1989]]
** This is sharp in the <span class="SpellE">focussing</span> case [[references.html#KnPoVe-p <span class="SpellE">KnPoVe</span>-p]], in the sense that the solution map is no longer uniformly continuous for s < 1/4.
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** This is sharp in the <span class="SpellE">focussing</span> case [[Bibliography#KnPoVe-p |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 [[references.html#Sch1998 Sch1998]]).
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*** 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 [[Bibliography#Sch1998|Sch1998]]).
*** Below 1/4 the solution map was known to not be C^3 in [[references.html#Bo1993b Bo1993b]], [[references.html#Bo1997 Bo1997]].
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*** Below 1/4 the solution map was known to not be C^3 in [[Bibliography#Bo1993b|Bo1993b]], [[Bibliography#Bo1997|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 [<span class="SpellE">CoKe</span>-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 [[references.html#Ka1983 Ka1983]]. Thus in L^2 one has global existence but no uniform continuity.
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** Global weak solutions in L^2 were constructed in [[Bibliography#Ka1983|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 [[references.html#GiTs1989 GiTs1989]]
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** Uniqueness is also known when the initial data lies in the weighted space <x>^{3/8} u_0 in L^2 [[Bibliography#GiTs1989|GiTs1989]]
** LWP has also been demonstrated when <xi>^s hat(u_0) lies in L^{r’} for 4/3 < r <= 2 and s >= ½ - 1/2r [Gr-p4]
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** 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 [[references.html#CoKeStTaTk-p2 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.
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* GWP in <span class="SpellE">H^s</span> for s > 1/4 [[Bibliography#CoKeStTaTk-p2 |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.
** Was proven for s>3/5 in [[references.html#FoLiPo1999 FoLiPo1999]]
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** Was proven for s>3/5 in [[Bibliography#FoLiPo1999|FoLiPo1999]]
** Is implicit for s >= 1 from [[references.html#KnPoVe1993 KnPoVe1993]]
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** Is implicit for s >= 1 from [[Bibliography#KnPoVe1993|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 [<span class="SpellE">CoKe</span>-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 [<span class="SpellE">BdmFsShp</span>-p]; the same approach also works on an interval [<span class="SpellE">BdmShp</span>-p].

Revision as of 03:13, 31 July 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 focussing 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 defocussing case in [CtCoTa-p], by a high-frequency approximation of mKdV by [schrodinger.html#Cubic NLS on R 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 CoKeStTkTa-p2, 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]