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On Existence and Uniqueness of Singular Solutions for Systems of Conservation Laws Kalisch, Henrik
Description
Existence and admissibility of delta-shock solutions is discussed for hyperbolic systems of conservation laws. One of the systems discussed is fully nonlinear, and does not admit a classical Lax-admissible solution to certain Riemann problems. By introducing complex-valued corrections in the framework of the weak asymptotic method, we show that a compressive delta-shock wave solution resolves such Riemann problems. By letting the approximation parameter tend to zero, the corrections become real valued and the resulting distributions fit into a generalized concept of singular solutions [V. G. Danilov and V. M. Shelkovich, Dynamics of propagation and interaction of delta-shock waves in hyperbolic systems, J. Differential Equations 211 (2005), 333-381]. In this framework, it can be shown that every 2x2 system of conservation laws admits delta-shock solutions. As an example, it is shown that the combination of discontinuous free-surface solutions and bottom step transitions naturally leads to singular solutions featuring Dirac delta distributions in the context of shallow-water flows.
Item Metadata
Title |
On Existence and Uniqueness of Singular Solutions for Systems of Conservation Laws
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Creator | |
Publisher |
Banff International Research Station for Mathematical Innovation and Discovery
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Date Issued |
2016-11-01T13:32
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Description |
Existence and admissibility of delta-shock solutions is discussed for hyperbolic systems of conservation laws. One of the systems discussed is fully nonlinear, and does not admit a classical Lax-admissible solution to certain Riemann problems.
By introducing complex-valued corrections in the framework of the weak asymptotic method, we show that a compressive delta-shock wave solution resolves such Riemann problems. By letting the approximation parameter tend to zero, the corrections become real valued and the resulting distributions fit into a generalized concept of singular solutions [V. G. Danilov and V. M. Shelkovich, Dynamics of propagation and interaction of delta-shock waves in hyperbolic systems, J. Differential Equations 211 (2005), 333-381]. In this framework, it can be shown that every 2x2 system of conservation laws admits delta-shock solutions.
As an example, it is shown that the combination of discontinuous free-surface solutions and bottom
step transitions naturally leads to singular solutions featuring Dirac delta distributions in the context of shallow-water flows.
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Extent |
37 minutes
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Subject | |
Type | |
File Format |
video/mp4
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Language |
eng
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Notes |
Author affiliation: University of Bergen
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Series | |
Date Available |
2017-05-03
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Provider |
Vancouver : University of British Columbia Library
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Rights |
Attribution-NonCommercial-NoDerivatives 4.0 International
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DOI |
10.14288/1.0347264
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URI | |
Affiliation | |
Peer Review Status |
Unreviewed
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Scholarly Level |
Faculty
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Rights URI | |
Aggregated Source Repository |
DSpace
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Rights
Attribution-NonCommercial-NoDerivatives 4.0 International