Delta Shocks and Vacuums to the Isentropic Euler Equations with the Flux Perturbation for van der Waals Gas

In this paper, we study the isentropic Euler equations with the flux perturbation for van der Waals gas, in which the density has both lower and upper bounds due to the introduction of the flux approximation and the molecular excluded volume. First, we solve the Riemann problem of this system and construct the Riemann solutions. Second, the formation mechanisms of delta shocks and vacuums are analyzed for the Riemann solutions as the pressure, the flux approximation, and the molecular excluded volume all vanish. Finally, some numerical simulations are demonstrated to verify the theoretical analysis.

Pressureless Magnetohydrodynamics System: Riemann Problem and Vanishing Magnetic Field Limit

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.