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

Kalisch, Henrik; Mitrović, Darko; Teyekpiti, Vincent

doi:10.1088/1361-6544/aae04b

Cited by 24 publications

(5 citation statements)

References 27 publications

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“…Although uniqueness was an unresolved issue for both of them. Recently the question of uniqueness is also settled in [14] by introducing some nonlinear change of variable in the flux function of the first equation of (1.4).…”

Section: General Euler Equations Of Compressible Fluid Flow Readsmentioning

confidence: 99%

Limiting behavior of scaled general Euler equations of compressible fluid flow

Sahoo

Sen

2020

Z. Angew. Math. Phys.

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The aim of this article is to study the limiting behavior of the solutions for the scaled generalized Euler equations of compressible fluid flow. When the initial data is of Riemann type, we showed the existence of solution which consists of shock waves and rarefaction waves and that the distributional limit of the solutions for this system converges to the solution of a non-strictly hyperbolic system, called one dimensional model for large scale structure formation of universe as the scaling parameter vanishes. An explicit entropy and entropy flux pair is also constructed for the particular flux function (Brio system) and it is shown that the solution constructed is entropy admissible. This is a continuation of our work [23].2010 Mathematics Subject Classification. 35L67, 35L65.

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Section: General Euler Equations Of Compressible Fluid Flow Readsmentioning

confidence: 99%

Limiting behavior of scaled general Euler equations of compressible fluid flow

Sahoo

Sen

2020

Z. Angew. Math. Phys.

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“…Recently, the approach of the adiabatic exponent limit has been well carried out for example in [15, 31, 34-36, 48, 49]. In addition, there are still some other very effective methods [9,16,17,23,25,27] to deal with the related problems of δ-shocks. Finally, it is necessary to point out that delta shock wave has also been found in other hyperbolic models, such as in the nonlinear chromatography [20], in the shallow water dynamics [18], in the thin film [28] and also in the porous media [13].…”

Section: Introductionmentioning

confidence: 99%

The adiabatic exponent limits of Riemann solutions for the extended macroscopic production model

Shan

Shen²,

Wei³

2022

Math. Model. Nat. Phenom.

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The exact Riemann solutions for the extended macroscopic production model with an adiabatic exponent are constructed in perfectly explicit forms. The asymptotic limit of Riemann solution consisting of 1-shock wave and 2-contact discontinuity tends to a delta shock solution for the pressureless gas dynamics model under the special over-compressive entropy condition as the adiabatic exponent drops to one. In contrast, the asymptotic limit of Riemann solution composed of 1-rarefaction wave and 2-contact discontinuity tends to the vacuum solution surrounded by two contact discontinuities by letting the adiabatic exponent tend to one, in which the state in the interior of the 1-rarefaction wave fan is developed into vacuum. The intrinsic phenomena of concentration and cavitation are identified and investigated carefully during this limiting process, which displays more complicated and completely different behavior compared with previous literature. In addition, some representative numerical calculations are also provided, which are in well agreement with our theoretical results.

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“…Physically, they can be used to express the concentration phenomenon. As for delta shocks, besides the papers cited above, also see [12][13][14][15][16][17][18][19]. The other situation is the vacuum state which is a state with ρ = 0.…”

Section: Introductionmentioning

confidence: 99%

Pressureless Magnetohydrodynamics System: Riemann Problem and Vanishing Magnetic Field Limit

Cheng

2020

Advances in Mathematical Physics

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This paper proposes the pressureless magnetohydrodynamics (MHD) system by neglecting the effect of pressure difference in the MHD system. Firstly, the Riemann problem for the pressureless MHD system is solved with five kinds of structures of solutions consisting of combinations of shock, rarefaction wave, contact discontinuity, and vacuum state. Secondly, the limit behavior of the obtained Riemann solutions as the magnetic field drops to zero is studied. It is shown that, as the magnetic field vanishes, the Riemann solutions of the pressureless MHD system just tend to the corresponding Riemann solutions of the Euler equations for pressureless fluids. The formation processes of delta shocks and vacuum states are clarified. For the delta shock, both the intermediate density and internal energy simultaneously develop delta measures.

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

Cited by 24 publications

References 27 publications

Limiting behavior of scaled general Euler equations of compressible fluid flow

Limiting behavior of scaled general Euler equations of compressible fluid flow

The adiabatic exponent limits of Riemann solutions for the extended macroscopic production model

Pressureless Magnetohydrodynamics System: Riemann Problem and Vanishing Magnetic Field Limit

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