# 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. | + | 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 [[ | + | * LWP in <span class="SpellE">H^s</span> for s >= 1/4 [[Bibliography#KnPoVe1993|KnPoVe1993]] |

− | ** Was shown for s>3/2 in [[ | + | ** Was shown for s>3/2 in [[Bibliography#GiTs1989|GiTs1989]] |

− | ** This is sharp in the <span class="SpellE">focussing</span> case [[ | + | ** 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 [[ | + | *** 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 [[ | + | *** 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 [[ | + | ** 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 [[ | + | ** 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^{ | + | ** 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 [[ | + | * 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 [[ | + | ** Was proven for s>3/5 in [[Bibliography#FoLiPo1999|FoLiPo1999]] |

− | ** Is implicit for s >= 1 from [[ | + | ** 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.
- 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]