On the rate of convergence and asymptotic profile of solutions to the viscous Burgers equation

In this paper we control the first moment of the initial approximations and obtain the order of convergence and the asymptotic profile of a general solution by two explicit "canonical" approximations: A diffusive N-wave and a diffusion wave solution. The order of convergence of both approximations is O(t^1/(2r)-3/2) in L^r norm, 1 ≤ r ≤ ∞, as t → ∞, which is faster than the well-known classical convergence order O(t^1/(2r)-1/2) for the inviscid Burgers equations case. A further comparison between the convergence rates of these two approximations and a discussion of the metastability phenomenon of the Burgers equation are also included. The method devised here allows us to obtain convergence up to any order by introducing new canonical solutions and controlling higher moments of the initial approximation.