Higher Order Asymptotics and Viscosity Method to Burgers Solutions

Abstract

The viscous Burgers equation ut +uux = νuxx is a nonlinear partial differential equation, named after the great physicist Johannes Martinus Burgers (1895-1981). We focused on the study of the large time asymptotic for solutions to the viscous Burgers equation and also to the adhesion model via heat equation. Using generalization of the truncated moment problem to a complex measure space, we construct asymptotic N-wave approximate solution to the heat equation subject to the initial data whose moments exist up to the order 2n + m and i-th order moment vanishes, for i = 0; 1; 2 : : : m − 1. We provide a different proof for a theorem given by Duoandikoetxea and Zuazua (1992), which plays a crucial role in error estimations. In addition to this we describe a simple way to construct an initial data in Schwartz class whose m moments are equal to the m moments of given initial data. Secondly, we focus on the Riemann problem for de-coupled system and obtain the weak solutions explicitly. It is to be noted here that real valued solution for the system exists in the case of rarefaction wave and the weak solution consist of δ- measures in the case of raising the speed of characteristics. Eventually, we consider inviscid Burgers equation with a forcing term, this is in fact the first equation in the de-coupled system, but with a general initial function u0(x) = o(jxj); as jxj ! 1: We then pick up an explicit solution from Satyanarayana et al. (2017) for the parabolic approximation of the hyperbolic partial differential equation using vanishing viscosity method, we construct weak solutions for the considered hyperbolic partial differential eq