A shallow-water system with vanishing buoyancy

Kalisch, Henrik; Teyekpiti, Vincent

doi:10.1080/00036811.2018.1546000

Cited by 7 publications

(4 citation statements)

References 33 publications

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“…In fact, in cases where the equations have singular solutions given in exact form, the weak asymptotic method can be shown to give the right solution (cf. [ 34 , 56 ] for example). However, there is not a firm proof that the weak asymptotic method works in every case.…”

Section: Discussionmentioning

confidence: 99%

On Existence and Admissibility of Singular Solutions for Systems of Conservation Laws

Kalisch

Mitrović

2022

Int. J. Appl. Comput. Math

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A weak notion of solution for systems of conservation laws in one dimension is put forward. In the framework introduced here, it can be shown that the Cauchy problem for any $$n\times n$$ n × n system of conservation laws has a solution. The solution concept is an extension of the notion of singular $$\delta $$ δ -shocks which have been used to provide solutions for Riemann problems in various systems, for example in cases where strict hyperbolicity or the genuine-nonlinearity condition are not satisfied, or in cases where initial conditions have large variation. We also introduce admissibility conditions which eliminate a wide range of unreasonable solutions. Finally, we provide an example from the shallow water system which justifies introduction of $$\delta $$ δ -distributions as a part of solutions to systems of conservation laws.

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Section: Discussionmentioning

confidence: 99%

On Existence and Admissibility of Singular Solutions for Systems of Conservation Laws

Kalisch

Mitrović

2022

Int. J. Appl. Comput. Math

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“…Formation of delta shocks for isentropic and nonisentropic fluids to the Euler equations when pressure vanishes has been studied by Chen and Liu [10,11]. The weak asymptotic method for delta shock wave was used in [12][13][14]. Yin and Sheng [15] discussed delta shock wave solution to relativistic Euler equations for polytropic gases.…”

Section: Introductionmentioning

confidence: 99%

“…which is the same as the Riemann solution of (1) and 3when u l \u r . Case II: If u r \u l \u r þ 2, from (14), the Riemann solution of (2) and (3) is J þ S. As g tends to zero, we obtain…”

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confidence: 99%

Stability of the Riemann solution for a 2 × 2 strictly hyperbolic system of conservation laws

2019

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In this work, we study the system of conservation laws that is strictly hyperbolic and whose Riemann solution contains delta shock waves as well as classical elementary waves. In order to study stability, we consider the linear approximation of flux functions with three parameters. The approximation does not affect the structure of Riemann solution. Furthermore, we prove that the solution of the Riemann problem for the approximated system converges to the solution of the original system when the perturbation parameter tends to zero.

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“…Some authors have defined theories of distribution products in order to incorporate the δ-distributions into the notion of weak solutions [4,10,23]. In other works, the need to multiply δ-distributions has been avoided either by working with integrated equations [9,13], or by making an appropriate definition of singular solutions [6]. In order to find admissibility conditions for such singular solutions, some authors have used the weak asymptotic method [5,6,21,22] or simply look for the limit of the vanishing viscosity approximation [15,24,25].…”

Section: Introductionmentioning

confidence: 99%

Existence and uniqueness of singular solutions for a conservation law arising in magnetohydrodynamics

2018

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The Brio system is a two-by-two system of conservation laws arising as a simplified model in ideal magnetohydrodynamics (MHD). The system has the formIt 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. It was argued in [8] that in order to solve the system, singular solutions containing Dirac masses along the shock waves might have to be used. Solutions of this type were exhibited in [11,23], but uniqueness was not obtained.In the current work, we introduce a nonlinear change of variables which makes it possible to solve the Riemann problem in the framework of the standard theory of conservation laws. In addition, we develop a criterion which leads to an admissibility condition for singular solutions of the original system, and it can be shown that admissible solutions are unique in the framework developed here.

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A shallow-water system with vanishing buoyancy

Cited by 7 publications

References 33 publications

On Existence and Admissibility of Singular Solutions for Systems of Conservation Laws

On Existence and Admissibility of Singular Solutions for Systems of Conservation Laws

Stability of the Riemann solution for a 2 × 2 strictly hyperbolic system of conservation laws

Existence and uniqueness of singular solutions for a conservation law arising in magnetohydrodynamics

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