# Difference between revisions of "Korteweg-de Vries equation on the half-line"

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* Some compatibility conditions between u_0 and h are needed. The higher the regularity, the more compatibility conditions are needed. If the initial data u_0 is in <span class="SpellE">H^s</span>, then by scaling heuristics the natural space for h is in H<span class="GramE">^{</span>(s+1)/3}. (Remember that time has dimensions ''length''^3). | * Some compatibility conditions between u_0 and h are needed. The higher the regularity, the more compatibility conditions are needed. If the initial data u_0 is in <span class="SpellE">H^s</span>, then by scaling heuristics the natural space for h is in H<span class="GramE">^{</span>(s+1)/3}. (Remember that time has dimensions ''length''^3). | ||

* LWP is known for initial data in <span class="SpellE">H^s</span> and boundary data in H<span class="GramE">^{</span>(s+1)/3} for s >= 0 [<span class="SpellE">CoKe</span>-p], assuming compatibility. The drift term may be omitted because of the time localization. | * LWP is known for initial data in <span class="SpellE">H^s</span> and boundary data in H<span class="GramE">^{</span>(s+1)/3} for s >= 0 [<span class="SpellE">CoKe</span>-p], assuming compatibility. The drift term may be omitted because of the time localization. | ||

− | ** For s > 3/4 this was proven in [[references | + | ** For s > 3/4 this was proven in [[references:BnSuZh-p <span class="SpellE">BnSuZh</span>-p]] (assuming that there is a drift term). |

− | ** Was proven for data in sufficiently weighted H^1 spaces in [[ | + | ** Was proven for data in sufficiently weighted H^1 spaces in [[Bibliography#Fa1983|Fa1983]]. |

** From the real line theory one might expect to lower this to -3/4, but there appear to be technical difficulties with this. | ** From the real line theory one might expect to lower this to -3/4, but there appear to be technical difficulties with this. | ||

* GWP is known for initial data in L^2 and boundary data in H<span class="GramE">^{</span>7/12}, assuming compatibility. | * GWP is known for initial data in L^2 and boundary data in H<span class="GramE">^{</span>7/12}, assuming compatibility. | ||

− | ** for initial data in H^1 and boundary data in H^{5/6}_loc this was proven in [[ | + | ** for initial data in H^1 and boundary data in H^{5/6}_loc this was proven in [[Bibliography#BnSuZh-p <span class="SpellE">|BnSuZh</span>-p]] |

− | ** Was proven for smooth data in [[ | + | ** Was proven for smooth data in [[Bibliography#BnWi1983|BnWi1983]] |

[[Category:Integrability]] | [[Category:Integrability]] | ||

[[Category:Equations]] | [[Category:Equations]] | ||

[[Category:Airy]] | [[Category:Airy]] |

## Revision as of 16:16, 31 July 2006

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

- The KdV Cauchy-boundary problem on the half-line is

The sign of u_{xxx} is important (it makes the influence of the boundary x=0 mostly negligible), the sign of u u_x is not. The drift term u_x appears naturally from the derivation of KdV from fluid mechanics. (On R, this drift term can be eliminated by a Gallilean transform, but this is not available on the half-line).

- Because one is restricted to the half-line, it becomes a little tricky to use the Fourier transform. One approach is to use the Fourier-Laplace transform instead.
- Some compatibility conditions between u_0 and h are needed. The higher the regularity, the more compatibility conditions are needed. If the initial data u_0 is in H^s, then by scaling heuristics the natural space for h is in H^{(s+1)/3}. (Remember that time has dimensions
*length*^3). - LWP is known for initial data in H^s and boundary data in H^{(s+1)/3} for s >= 0 [CoKe-p], assuming compatibility. The drift term may be omitted because of the time localization.
- For s > 3/4 this was proven in [[references:BnSuZh-p BnSuZh-p]] (assuming that there is a drift term).
- Was proven for data in sufficiently weighted H^1 spaces in Fa1983.
- From the real line theory one might expect to lower this to -3/4, but there appear to be technical difficulties with this.

- GWP is known for initial data in L^2 and boundary data in H^{7/12}, assuming compatibility.
- for initial data in H^1 and boundary data in H^{5/6}_loc this was proven in [[Bibliography#BnSuZh-p |BnSuZh-p]]
- Was proven for smooth data in BnWi1983