Hyperbolic conservation laws with large initial data. Is the Cauchy problem well-posed?
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In this paper, the Glimm estimate for a strictly hyperbolic system of two conservation laws is violated, which is the only known example of such a theorem being violated.Abstract:
We present an example in which the Glimm estimate for a strictly hyperbolic system of two conservation laws is violated.read more
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Singular solutions of a fully nonlinear 2x2 system of conservation laws
Henrik Kalisch,Darko Mitrović +1 more
TL;DR: In this article, the existence and admissibility of a compressive δ-shock-type solution to Riemannian problems is discussed. And the authors show that every $2\times 2$ system of conservation laws admits such a solution in the weak asymptotic method.
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Existence and Uniqueness of Singular Solutions for a Conservation Law Arising in Magnetohydrodynamics
TL;DR: The Brio system is a two-by-two system of conservation laws arising as a simplified model in ideal magnetohydrodynamics (MHD), and it was found in previous works that the standard theory of hyperbolic conservation laws does not apply to this system since the characteristic fields are not genuinely nonlinear on the set $v=0, as a consequence, certain Riemann problems have no weak solutions in the traditional class of functions of bounded variation as discussed by the authors.
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Existence and uniqueness of singular solutions for a conservation law arising in magnetohydrodynamics
TL;DR: In this paper, the Brio system is solved in the standard theory of conservation laws and a criterion which leads to an admissibility condition for singular solutions of the original system is developed.
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Admissibility conditions for Riemann data in shallow water theory
Martin Paulsen,Henrik Kalisch +1 more
TL;DR: In this article, it is shown that if certain physical assumptions on the disposition of the Riemann data are made, then dry states can be avoided in the solution of the problem.
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On Existence and Admissibility of Singular Solutions for Systems of Conservation Laws
Henrik Kalisch,Darko Mitrović +1 more
TL;DR: In this paper , a weak notion of solution for systems of conservation laws in one dimension is put forward, and it can be shown that the Cauchy problem for any $$n\times n$$ has a solution.
References
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Book
Shock Waves and Reaction-Diffusion Equations
TL;DR: In this paper, the basics of hyperbolic conservation laws and the theory of systems of reaction-diffusion equations, including the generalized Morse theory as developed by Charles Conley, are presented in a way accessible to a wider audience than just mathematicians.
Book
Hyberbolic Conservation Laws in Continuum Physics
TL;DR: In this paper, the authors present a masterly exposition and an encyclopedic presentation of the theory of hyperbolic conservation laws, with a focus on balance laws with dissipative source, modeling relaxation phenomena.
Journal ArticleDOI
Hyperbolic Systems of Conservation Laws
TL;DR: In this article, the authors present a survey of recent advances in the mathematical theory of hyperbolic systems of conservation laws in one space dimension and present some of the latest results on uniqueness and stability of entropy weak solutions.
Book
Front Tracking for Hyperbolic Conservation Laws
Helge Holden,Nils Henrik Risebro +1 more
TL;DR: In this paper, the authors presented the solution of the Riemann problem for the 3 3 Euler equations of gas dynamics, and the main existence theorem in Chapt. 4, Theorem 4.3, now resembles the one-dimensional case.