Generalized regularized long waves equations with white noise dispersion

Abstract

In this article, we address the generalized BBM equation with white noise dispersion which reads \begin{equation*} du-du_{xx}+u_x \circ dW+ u^pu_xdt=0, \end{equation*} in { Stratonovich} formulation, where $W(t)$ is a standard real valued Brownian motion. It is equivalent to, in Ito's formulation, \begin{equation*} du+(1-\Delta)^{-1}u_x dW- \frac{1}{2}(1-\Delta)^{-2}u_{xx}dt+ (1-\Delta)^{-1}u^pu_xdt=0. \end{equation*} The global well posedeness for the initial value problem is shown for any $p\geq 1$. Furthermore, due to the particular structure of the Stratonovich product, the $H^1_x$ 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 $L^\infty_x$ norm of the solutions decay to zero at $O(t^{-\frac16})$, 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 $L^1_x\cap H^4_x$. Numerical simulations are presented to demonstrate the theoretical results obtained on decay rates are sharp.