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

Part of: Representations of solutions Hyperbolic equations and systems Distributions, generalized functions, distribution spaces

Published online by Cambridge University Press:  19 July 2019

C. O. R. Sarrico  and
A. Paiva
C. O. R. Sarrico
Affiliation:
CMAFCIO, Faculdade de Ciências da Universidade de Lisboa, Campo Grande, 1749-016 Lisboa, Portugal (corsarrico@gmail.com; paiva@cii.fc.ul.pt)
A. Paiva
Affiliation:
CMAFCIO, Faculdade de Ciências da Universidade de Lisboa, Campo Grande, 1749-016 Lisboa, Portugal (corsarrico@gmail.com; paiva@cii.fc.ul.pt)
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Abstract

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.

Keywords

Products of distributionsconservation lawstriangular systemgeneralized pressureless gas dynamics systemδ-shock wavesδ′-shock wavesCauchy problem

MSC classification

Primary: 46F10: Operations with distributions
Secondary: 35D99: None of the above, but in this section 35L67: Shocks and singularities
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 6 , December 2020 , pp. 2757 - 2775
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Copyright
Copyright © 2019 The Royal Society of Edinburgh

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