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.

No abstract

The Riemann problem for the anti-Chaplygin pressure Aw–Rascle model with a Coulomb-like friction term is considered. With the use of the substitution of variables, the Riemann solutions with two or three kinds of different structures involving the delta shock wave in two cases are constructed. The delta shock wave may be used to explain the serious traffic jam. The position, strength, and propagation speed of the delta shock wave are obtained by solving the generalized Rankine–Hugoniot relation under an entropy condition. Moreover, the results show that all waves including the contact discontinuity, rarefaction wave, shock wave, and delta shock wave are bent into parabolic shapes and the Riemann solutions are no longer self-similar under the influence of the Coulomb-like friction term.

This paper studies the system of conservation laws of mass, momentum, and energy in zero-pressure gas dynamics. By the vanishing viscosity method, the stability of the solutions involving delta shock wave with Dirac delta functions developing in both state variables is established.

No abstract