Distributional products and global solutions for nonconservative inviscid Burgers equation

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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(∂u2/∂x) in distributional sense. In this paper we will treat non-conservative 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.




Sarrico, C. O. R. (2003). Distributional products and global solutions for nonconservative inviscid Burgers equation. Journal of Mathematical Analysis and Applications, 281(2), 641–656. https://doi.org/10.1016/S0022-247X(03)00187-2

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