View source for Korteweg-de Vries equation on the half-line

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

* The <span class="SpellE">KdV</span> Cauchy-boundary problem on the half-line is

<center><span class="SpellE">u_t</span> + u<span class="GramE">_{</span>xxx} + <span class="SpellE">u_x</span> + u <span class="SpellE">u_x</span> = 0; u(x,0) = u_0(x); u(0,t) = h(t)</center>

The sign of u<span class="GramE">_{</span>xxx} is important (it makes the influence of the boundary x=0 mostly negligible), the sign of u <span class="SpellE">u_x</span> is not. The drift term <span class="SpellE">u_x</span> appears naturally from the derivation of <span class="SpellE">KdV</span> from fluid mechanics. (On R, this drift term can be eliminated by a <span class="SpellE">Gallilean</span> 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-<span class="SpellE">Laplace</span> 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 <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.
** For s > 3/4 this was proven in [[BnSuZh-p]] (assuming that there is no 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<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 [[BnSuZh-p]]
** Was proven for smooth data in [[BnWi1983]]

[[Category:Integrability]]
[[Category:Equations]]
[[Category:Airy]]