Distributional products and global solutions for nonconservative inviscid Burgers equation

Abstract

Burgers equation for inviscid fluids is a simplified case of Navier–Stokes equation which corresponds to Euler equation for ideal fluids. Thus, from a variational viewpoint, Burgers equation appears naturally in its nonconservative form. In this form, a consistent concept of a weak solution cannot be formulated because the classical distribution theory has no products which account for the term u( ∂u/ ∂x). This leads several authors to substitute Burgers equation by the so-called conservative form, where one has 1 2 (∂u 2/∂x) in distributional sense. In this paper we will treat nonconservative inviscid Burgers equation and study it with the help of our theory of products; also, the relationship with the conservative Burgers equation is considered. In particular, we will be able to exhibit a Dirac- δ travelling soliton solution in the sense of global α-solution. Applying our concepts, solutions which are functions with jump discontinuities can also be obtained and a jump condition is derived. When we replace the concept of global α-solution by the concept of global strong solution, this jump condition coincides with the well-known Rankine–Hugoniot jump condition for the conservative Burgers equation. For travelling waves functions these concepts are all equivalent.