Explicit Solution of a Non-strictly Hyperbolic System with Discontinuous Flux Using a Scaling Argument

In this paper, we consider the initial value problem for a 2 × 2 system of non-strictly hyperbolic conservation laws. The first equation is a convex conservation law whose flux has linear growth at infinity and has an asymptotic limit under a scaling. The second equation is linear and exhibits measu...

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Bibliographic Details
Published inDifferential equations and dynamical systems Vol. 32; no. 4; pp. 1175 - 1192
Main Authors Das, Abhishek, Joseph, K. T.
Format Journal Article
LanguageEnglish
Published New Delhi Springer India 01.10.2024
Springer Nature B.V
Subjects
Asymptotic methods
Asymptotic properties
Boundary value problems
Computer Science
Conservation laws
Engineering
Hyperbolic systems
Mathematics
Mathematics and Statistics
Original Research
Scaling
Scaling properties
Primary 35L65
Lax equation
Secondary 35A22, 35B40, 35D30
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ISSN0971-3514
0974-6870
DOI10.1007/s12591-022-00620-z

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Summary:In this paper, we consider the initial value problem for a 2 × 2 system of non-strictly hyperbolic conservation laws. The first equation is a convex conservation law whose flux has linear growth at infinity and has an asymptotic limit under a scaling. The second equation is linear and exhibits measure-valued solution even if the initial data is smooth with compact support. We use a scaling argument to derive explicit formula of solution for the system with the asymptotic flux function where the second equation becomes linear with discontinuous coefficient. We also study properties of solution when the initial data is periodic with zero mean over the period. Our theory is illustrated using the Lax equation.
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ISSN:0971-3514
0974-6870
DOI:10.1007/s12591-022-00620-z