Limit set of trajectories of the coupled viscous Burgers' equations

Abstract

In this letter, a coupled system of viscous Burgers' equations with zero Dirichlet boundary conditions and appropriate initial data is considered. For the well-known single viscous Burgers' equation with zero Dirichlet boundary conditions, the zero equilibrium is the unique global exponential point attractor. A similar property is shown for the coupled Burgers' equations, i.e., trajectories starting with initial data which is not too large, approach the zero equilibrium as time goes to infinity. This “approaching” or convergence is not necessarily exponentially fast, unlike the single viscous Burgers' equation.