Products of Distributions,Conservation Laws and the Propagation of δ'-Shock Waves

This paper contains a study of propagation of singular travelling waves u(x,t) for conservation laws ut + [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 ut +(u2/2) x = 0,the diffusionless Burgers-Fischer equation ut + a(u2/2) x = ru(1-u/k) with a,r,k being positive numbers,Leveque and Yee equation ut + ux = μu(1-u)(u-1/2) 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.

This paper contains a study of propagation of singular travelling waves u（x, t） for conservation laws ut ＋ [Ф（u）]x = ψ（u）, where Ф, ψ are entire functions taking real values on the real axis. Conditions for the propagation of wave profiles β ＋ mδ and β ＋ mδt are presented （β is a real continuous function, m ≠ 0 is a real number and δ＇ is the derivative of the Dirac measure 5）. These results are obtained with a consistent concept of solution based on our theory of distributional products. Burgers equation ut ＋ （u2/2）x = 0, the iffusionless Burgers-Fischer equation ut ＋ a（u2/2）x = ru（1 - u/k） with a, r, k being positive numbers, Leveque and Yee equation ut ＋ ux = μx（1 - u）（u - u/k） 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.