The initial-value problem for the Korteweg-de Vries equation

Abstract

For the Korteweg-de Vries equation u$_{t}$ + u$_{x}$ + uu$_{x}$ + u$_{xxx}$ = 0, existence, uniqueness, regularity and continuous dependence results are established for both the pure initial-value problem (posed on -$\infty $ < x < $\infty $) and the periodic initial-value problem (posed on 0 $\leq $ x $\leq $ l with periodic initial data). The results are sharper than those obtained previously in that the solutions provided have the same number of L$_{2}$-derivatives as the initial data and these derivatives depend continuously on time, as elements of L$_{2}$. The same equation with dissipative and forcing terms added is also examined. A by-product of the methods used is an exact relation between solutions of the Korteweg-de Vries equation and solutions of an alternative model equation recently studied by Benjamin, Bona & Mahony (1972). It is proven that in the long-wave limit under which these equations are generally derived, the solutions of the two models posed for the same initial data are the same. In the process of carrying out this programme, new results are obtained for the latter model equation.