Weak solutions for Euler systems with non-local interactions

We consider several modifications of the Euler system of fluid dynamics, including its pressureless variant driven by non‐local interaction repulsive–attractive and alignment forces in the space dimension =2,3. These models arise in the study of self‐organization in collective behavior modeling of animals and crowds. We adapt the method of convex integration to show the existence of infinitely many global‐in‐time weak solutions for any bounded initial data. Then we consider the class of dissipative solutions satisfying, in addition, the associated global energy balance (inequality). We identify a large set of initial data for which the problem admits infinitely many dissipative weak solutions. Finally, we establish a weak–strong uniqueness principle for the pressure‐driven Euler system with non‐local interaction terms as well as for the pressureless system with Newtonian interaction.