On the Lagrangian Trajectories for the One-Dimensional Euler Alignment Model without Vacuum Velocity

Abstract

A well-known result of Carrillo, Choi, Tadmor, and Tan [1] states that the 1D Euler Alignment model with smooth interaction kernels possesses a "critical threshold" criterion for the global existence or finite-time blowup of solutions, depending on the global nonnegativity (or lack thereof) of the quantity e 0 = ∂ x u 0 + φ * ρ 0. In this note, we rewrite the 1D Euler Alignment model as a first-order system for the particle trajectories in terms of a certain primitive ψ 0 of e 0 ; using the resulting structure, we give a complete characterization of global-in-time existence versus finite-time blowup of regular solutions that does not require a velocity to be defined in the vacuum. We also prove certain upper and lower bounds on the separation of particle trajectories, valid for smooth and weakly singular kernels, and we use them to weaken the hypotheses of Tan [25] sufficient for the global-in-time existence of a solution in the weakly singular case, when the order of the singularity lies in the range s ∈ (0, 1 2) and the initial data is critical.

Sticky particle Cucker-Smale dynamics and the entropic selection principle for the 1D Euler-alignment system

“…Our first observation is that the occurrence of collisions is strongly connected to the monotonicity of the values (ψ i ) N i=1 in i. The proof is similar in spirit to the work [44] by the first author, which is formulated at the hydrodynamic rather than the discrete level.…”

“…In what follows, we prove several properties of the quantities e i,j , including an analog of (44). We shall keep in mind the trivial bound…”

“…In this case a slightly different critical threshold condition was obtained in [56] (c.f. also [44]). The global behavior is similar to the case when φ is bounded.…”

“…In the presence of vacuum ρ 0 (x) = 0, the critical threshold condition is inaccessible from physical initial data (ρ 0 , P 0 ). The first author [44] works with an antiderivative ψ of e, defined by (8) ψ(x, t) = u(x, t) + Φ * ρ(x, t),…”

Geometric Structure of Mass Concentration Sets for Pressureless Euler Alignment Systems

“…What stops the unidirectional dynamics from being completely one-dimensional is the convolution term in (8), which depends on values of the density in all stratification layers. Wellposedness for the system (8)- (10) for solutions satisfying e 0 ≥ 0 is presented in [9]. One explanation for the prominent role of e 0 in the 1D wellposedness theory is that the quantity (11) β α e 0 (γ)dγ controls the long-time separation of the trajectories X(α, t) and X(β, t) originating at α, β.…”