Korteweg-de Vries equation on the half-line: Difference between revisions
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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 [[ | ** For s > 3/4 this was proven in [[Bibliography#BnSuZh-p |BnSuZh-p]] (assuming that there is no drift term). | ||
** Was proven for data in sufficiently weighted H^1 spaces in [[Bibliography#Fa1983|Fa1983]]. | ** 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 [[Bibliography#BnSuZh-p | ** 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 [[Bibliography#BnWi1983|BnWi1983]] | ** Was proven for smooth data in [[Bibliography#BnWi1983|BnWi1983]] | ||
Revision as of 16:21, 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.
- GWP is known for initial data in L^2 and boundary data in H^{7/12}, assuming compatibility.