Korteweg-de Vries equation on the half-line

From DispersiveWiki

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 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^{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