Distributions as initial values in a triangular hyperbolic system of conservation laws

Abstract

AbstractThe present paper concerns the systemut + [ϕ(u)]x = 0,vt + [ψ(u)v]x = 0 having distributions as initial conditions. Under certain conditions, and supposingϕ,ψ: ℝ → ℝ functions, we explicitly solve this Cauchy problem within a convenient space of distributionsu,v. For this purpose, a consistent extension of the classical solution concept defined in the setting of a distributional product (not constructed by approximation processes) is used. Shock waves,δ-shock waves, and also waves defined by distributions that are not measures are presented explicitly as examples. This study is carried out without assuming classical results about conservation laws. For reader's convenience, a brief survey of the distributional product is also included.