Unidirectional Entropic Solutions of the Pressureless Euler Alignment System
We develop a global wellposedness theory for weak solutions of the pressureless Euler Alignment system with measure-valued density $ρ$, bounded and unidirectional velocity $\mathbf{u} = (u,0, \ldots, 0)$, and a communication protocol $ϕ$ that may be bounded or weakly singular. This appears to be the first such theory that admits shocks outside one space dimension. We recast the system as a family of nonlocally coupled scalar balance laws---one for each horizontal slice of $\mathbb{R}^d$---and establish existence, uniqueness, and stability of entropy solutions of the reformulated system, before translating back to the level of $ρ$ and $u$. This grants us access to certain key one-dimensional tools in the direction of the flow. The analysis in the directions transverse to the flow, however, still presents substantial challenges: The horizontal slices are coupled and therefore cannot be treated independently, and two solutions need not distribute their mass over horizontal slices in the same way, so that there is no canonical way to compare them slicewise. Accordingly, our analysis is centered around the nonlocal alignment force, and our framework leverages optimal couplings between the projections onto $\mathbb{R}^{d-1}$ of the density profiles under consideration. We construct our solutions as limits of atomic density and momentum profiles whose atoms follow sticky particle Cucker--Smale dynamics, and under additional regularity assumptions, we obtain quantitative rates of convergence. Finally, our investigation of the long-time behavior of unidirectional solutions highlights the role of lateral communication in driving the system toward a limiting density profile: We prove that flocking occurs, at a rate independent of the number of agents, under assumptions on $ϕ$ that bound its size from below only outside a cylindrical neighborhood of the axis parallel to the flow.