Periodic Solutions to the Steady Viscous Burgers Equation: A Constructive Example

Abstract

In this paper, we consider the periodic problem to steady viscous Burgers equation with an periodic and external force. Our main aim is to construct periodic solutions to this model which are uniformly bounded with respect to the viscosity via a direct Fourier series approach. Firstly, as an example, starting from a special solution of the non-viscous Burgers equation with a specific force, we solve the approximate solutions to the viscous Burgers equation in an explicit way. Secondly, we construct the solution to the periodic problem of the viscous Burgers equation with the external force by solving the initial value problem to a second ordinary differential equations, which is uniformly bounded with respect to the viscosity. Compared with Jauslin-Kreiss-Moser's result, our approach provides an explicit constructive procedure for the periodic solutions and yields a sharper convergence rate (up to higher order) when the viscosity vanishes.