Exponential Stability of the Flow for a Generalized Burgers Equation on a Circle

The paper deals with the problem of stability for the flow of the 1D Burgers equation on a circle. Using some ideas from the theory of positivity preserving semigroups, we establish the strong contraction in the L¹ norm. As a consequence, it is proved that the equation with a bounded external force possesses a unique bounded solution on \({\mathbb{R}}\) $\mathbb{R}$ , which is exponentially stable in H¹ as t → +∞. In the case of a random external force, we show that the difference between two trajectories goes to zero with probability 1.