An initial-boundary-value problem that approximate the quarter-plane problem for the Korteweg–de Vries equation

In this paper, we continue the study of the initial-boundary-value prob- lem for the Korteweg-de Vries equation that has been initiated in (9). We obtain global smoothing eects that are uniform with respect to the size of the interval. This allows us to show that the solution of the boundary value problem converges, as the size of the interval converges to infinity, towards the solution of the quarter-plane problem. We also propose a simple finite dierents scheme for the problem on (0 ,1) and prove its stability.

[1]  D. Korteweg,et al.  XLI. On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves , 1895 .

[2]  Ragnar Winther,et al.  The Korteweg–de Vries Equation, Posed in a Quarter-Plane , 1983 .

[3]  J. Bourgain,et al.  Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations , 1993 .

[4]  Jean-Michel Ghidaglia,et al.  An initial-boundary value problem for the Korteweg-de Vries equation posed on a finite interval , 2001, Advances in Differential Equations.

[5]  F. Smith,et al.  Conservative, high-order numerical schemes for the generalized Korteweg—de Vries equation , 1995, Philosophical Transactions of the Royal Society of London. Series A: Physical and Engineering Sciences.

[6]  T. Colin,et al.  Un problème mixte pour l’équation de Korteweg-de Vries sur un intervalle borné , 1997 .

[7]  R. Temam.  Sur la résolution exacte et approchée d'un problème hyperbolique non linéaire de T. Carleman , 1969 .

[8]  J. Bona,et al.  The initial-value problem for the Korteweg-de Vries equation , 1975, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[9]  Luis Vega,et al.  A bilinear estimate with applications to the KdV equation , 1996 .

[10]  J. Bona,et al.  The Korteweg-de Vries equation in a quarter plane, continuous dependence results , 1989, Differential and Integral Equations.

[11]  J. Bourgain,et al.  Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations , 1993 .