Generalized Regularized Long Waves Equations with White Noise Dispersion

In this article, we address the generalized BBM equation with white noise dispersion which reads du− duxx + ux ◦ dW + uuxdt = 0, in Stratonovich formulation, where W (t) is a standard real valued Brownian motion. It is equivalent to, in Ito’s formulation, du + (1−∆)uxdW − 1 2 (1−∆)uxxdt + (1−∆)uuxdt = 0. The global well posedeness for the initial value problem is shown for any p ≥ 1. Furthermore, due to the particular structure of the Stratonovich product, the H x norm of the solutions is proved to be conserved with respect to time. It is also proved that for the linearized problem, the expectation of the Lx norm of the solutions decay to zero at O(t − 1 6 ), which is a slower rate than the solutions of the corresponding deterministic equation. A similar result is then proved for the nonlinear problem by assuming that p > 8 and that the initial data is small in Lx∩H x. Numerical simulations are presented to demonstrate the theoretical results obtained on decay rates are sharp.