The stochastic Korteweg-de Vries equation in \(L^2(\mathbb{R})\) (Q1288197)

The author presents two results of global existence and uniqueness of the Cauchy problem concerning the Korteweg-de Vries equation forced by random term of white noise type in the functional framework of weighted spaces on \(L^2(\mathbb{R})\). This work is motivated both by physical experiments where a forced generation of weakly nonlinear waves by localized disturbances was observed [see \textit{T. R. Akylas}, J. Fluid. Mech. 141, 455-466 (1984; Zbl 0551.76018) and \textit{S. J. Lee}, \textit{G. T. Yates} and \textit{T. Y. Wu}, J. Fluid. Mech. 199, 569-593 (1989)] and by a result of uniqueness in weighted space in the homogeneous case [see \textit{J. Ginibre} and \textit{Y. Tsutsumi}, SIAM J. Math. Anal. 20, No. 6, 1388-1425 (1989; Zbl 0702.35224), and \textit{S. N. Krushkov} and \textit{A. V. Faminskij}, Math. USSR, Sb. 48, 391-421 (1984; Zbl 0549.35104)].