OxPDE Lunchtime Seminar

I will discuss the multi-dimensional Euler–alignment system with a matrix-valued communication kernel, which is motivated by models of anticipation dynamics in collective behaviour. A key feature of this system is its formal equivalence to a nonlocal variant of the Aw–Rascle–Zhang (ARZ) traffic model, in which the desired velocity is modified by a nonlocal gradient interaction. The global-in-time existence of measure solutions to both formulations,  can be obtained via a single degenerate pressureless Navier–Stokes approximation. I will also discuss a weak–strong uniqueness principle adapted to the pressureless setting and to nonlocal alignment forces. As a consequence of these results we can rigorously justify the formal correspondence between the nonlocal ARZ and Euler–alignment models: they arise from the same inviscid limit, and the weak–strong uniqueness property ensures that, whenever a classical solution exists, both formulations coincide with it.