Exponential stability of the flow for a generalized Burgers equation on a circle

Abstract

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 \(L^1\) norm. As a consequence, it is proved that the equation with a bounded external force possesses a unique bounded solution on\(R\), which is exponentially stable in\(H^1\) as \(t\to+\infty\). In the case of a random external force, we show that the difference between two trajectories goes to zero with probability\(1\).