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 12
(∂u2/∂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.
2003 Elsevier Science (USA). All rights reserved
