Bulletin of the
Korean Mathematical Society BKMS

We study the Riemann problem for the pressureless Euler system with the source term depending on the time. By means of the variable substitution, two kinds of Riemann solutions including delta-shock and vacuum are constructed. The generalized Rankine-Hugoniot relation and entropy condition of the delta-shock are clarified. Because of the source term, the Riemann solutions are non-self-similar. Moreover, we propose a time-dependent viscous system to show all of the existence, uniqueness and stability of solutions involving the delta-shock by the vanishing viscosity method.

Keywords: Pressureless Euler system, source term, non-self-similar solutions, delta-shock, vanishing viscosity method

MSC numbers: 35L65, 35L67, 35B30

Supported by: This paper is supported by the Ph.D. Foundation of Henan University of Engineering (D2022028), the Science and Technology Research Program of Education Department of Henan Province (22A110008).