Products of distributions, conservation laws and the propagation of δ′-shock waves

Abstract

This paper contains a study of propagation of singular travelling waves u(x, t) for conservation laws u _t +[ϕ(u)] _x = ψ(u), where ϕ, ψ are entire functions taking real values on the real axis. Conditions for the propagation of wave profiles β + mδ and β + mδ′ are presented (β is a real continuous function, m ≠ 0 is a real number and δ′ is the derivative of the Dirac measure δ). These results are obtained with a consistent concept of solution based on our theory of distributional products. Burgers equation $u_t + \left( {\tfrac{{u^2 }} {2}} \right)_x = 0$, the diffusionless Burgers-Fischer equation $u_t + a\left( {\tfrac{{u^2 }} {2}} \right)_x = ru\left( {1 - \tfrac{u} {k}} \right)$ with a, r, k being positive numbers, Leveque and Yee equation $u_t + u_x = uu\left( {1 - u} \right)\left( {u - \tfrac{1} {2}} \right)$ with μ ≠ 0, and some other examples are studied within such a setting. A “tool box” survey of the distributional products is also included for the sake of completeness.