Non-uniqueness of admissible weak solutions to compressible Euler systems with source terms

• Prove the non-uniqueness of admissible weak solutions to the Euler system with sources.
• Prove the existence of multiple admissible weak solutions to the Euler system with sources with only finite states.
• Construct localized plane wave perturbations for the subsolution of the Euler system with sources.

We consider admissible weak solutions to the compressible Euler system with source terms, which include rotating shallow water system and the Euler system with damping as special examples. In the case of anti-symmetric sources such as rotations, for general piecewise Lipschitz initial densities and some suitably constructed initial momentum, we obtain infinitely many global admissible weak solutions. Furthermore, we construct a class of finite-states admissible weak solutions to the Euler system with anti-symmetric sources. Under the additional smallness assumption on the initial densities, we also obtain multiple global-in-time admissible weak solutions for more general sources including damping. The basic framework are based on the convex integration method developed by De Lellis and Székelyhidi [13] and [14] for the Euler system. One of the main ingredients of this paper is the construction of specified localized plane wave perturbations which are compatible with a given source term.