Weak solutions for Euler systems with non-local interactions

Abstract

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 N = 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.