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

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** 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 <math><\xi> ^s \hat{u_0}</math> lies in <math>L^{r}</math> for 4/3 < r <= 2 and s >= ½ - 1/2r [[Gr-p4]]
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** LWP has also been demonstrated when <math><\xi> ^s \hat{u_0}</math> lies in <math>L^{r/(r-1)}</math> for 4/3 < r <= 2 and s >= ½ - 1/2r [[Gr-p4]]
 
* 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.
 
* 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]]

Revision as of 00:45, 17 March 2007

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 KnPoVe2001, 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 CoKn-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 lies in 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 CoKn-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