Burgers system driven by a periodic stochastic flow

Abstract

Burgers System (BS) is Navier-Stokes system without pressure and incompressibility. It is one of the most popular models of hydrodynamics and has a lot of applications. In this paper, we consider BS driven by a potential force whose potential is a periodic stochastic flow. If x = (x ₁,... x _n) is the vector of coordinates and u = (u ₁,..., u„) is the velocity vector then the n-dimensional BS takes the form

$$\frac{\partial u}{\partial t}+(u,\nabla )u=\mu \Delta u+\nabla \dot{B}(x,t)$$

(1)

or in the coordinate form

$$\frac{\partial {{u}_{i}}}{\partial t}+\sum\limits_{k=1}^{n}{\frac{\partial {{u}_{i}}}{\partial {{x}_{k}}}}\cdot {{u}_{k}}=\mu \Delta {{u}_{i}}+\frac{\partial }{\partial {{x}_{i}}}\dot{B}(x,t)$$

(2)

.