# Korteweg-de Vries equation on the half-line

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Jump to navigationJump to searchThe 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