Asymptotic behavior of solutions to the viscous Burgers equation

Abstract

We study the asymptotic behavior of solutions to the viscous Burgers equation by presenting a new asymptotic approximate solution. This approximate solution, called a diffusion wave approximate solution to the viscous Burgers equation of $k$-th order, is expanded in terms of the initial moments up to $k$-th order. Moreover, the spatial and time shifts are introduced into the leading order term to capture precisely the effect of the initial data on the long-time behavior of the actual solution. We also show the optimal convergence order in $L^p$-norm, $1\leq p\leq \infty$, of the diffusion wave approximate solution of $k$-th order. These results allow us to obtain the convergence of any higher order in $L^p$-norm by taking such a diffusion wave approximate solution with order $k$ large enough.