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.html#BnSuZh-p <span class="SpellE">BnSuZh</span>-p]] (assuming that there is a drift term).
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** 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 [[references.html#Fa1983 Fa1983]].
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** 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 [[references.html#BnSuZh-p <span class="SpellE">BnSuZh</span>-p]]
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** 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 [[references.html#BnWi1983 BnWi1983]]
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** 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
u_t + u_{xxx} + u_x + u u_x = 0; u(x,0) = u_0(x); u(0,t) = h(t)

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