Pressureless Magnetohydrodynamics System: Riemann Problem and Vanishing Magnetic Field Limit

Abstract

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.

Delta shock wave as limits of vanishing viscosity for zero‐pressure gas dynamics with energy conservation law

“…Li et al. [11] analyzed the formation of delta shock waves and vacuum states in the Riemann solutions of the pressureless magnetohydrodynamics system with () as magnetic field vanishes. As regards the delta shock wave, it is well known as a kind of discontinuity, on which at least one of the state variables may develop an extreme concentration in the form of a weighted Dirac delta function with the discontinuity as its support.…”

“…Comparing with the transport equations, a distinctive feature for the system (1.1) is that the Dirac delta functions develop in two state variables 𝜌 and 𝐻 simultaneously. Li et al [11] analyzed the formation of delta shock waves and vacuum states in the Riemann solutions of the pressureless magnetohydrodynamics system with (1.2) as magnetic field vanishes.…”