Non-uniqueness of admissible weak solutions to the two-dimensional pressureless Euler system

Abstract

We study Riemann problem for the two-dimensional (2D) pressureless Euler system with planar Riemann initial data. It is proved that there exist infinitely many bounded admissible weak solutions to the 2D Riemann problem by the method of convex integration. Meanwhile, the corresponding one-dimensional (1D) Riemann problem admits a unique measure-valued solution (so-called δ-shock) under the Oleĭnik's entropy condition and an additional energy condition, which implies the non-existence of 1D bounded admissible weak solutions with energy condition (cf. [19]).