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δ′-shock waves

Proceedings of the Royal Society of Edinburgh Section A: Mathematics (1)

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Distributions as initial values in a triangular hyperbolic system of conservation laws
Part of
C. O. R. Sarrico, A. Paiva
Journal:

Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 150 / Issue 6 / December 2020

Published online by Cambridge University Press:

19 July 2019, pp. 2757-2775

Print publication:

December 2020
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The present paper concerns the system ut + [ϕ(u)]x = 0, vt + [ψ(u)v]x = 0 having distributions as initial conditions. Under certain conditions, and supposing ϕ, ψ: ℝ → ℝ functions, we explicitly solve this Cauchy problem within a convenient space of distributions u,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.