The global Cauchy problem for compressible Euler equations with a nonlocal dissipation

Abstract

This paper studies the global existence and uniqueness of strong solutions and its largetime behavior for the compressible isothermal Euler equations with a nonlocal dissipation. The system is rigorously derived from the kinetic Cucker-Smale flocking equation with strong local alignment forces and diffusions through the hydrodynamic limit based on the relative entropy argument. In a perturbation framework, we establish the global existence of a unique strong solution for the system under suitable smallness and regularity assumptions on the initial data. We also provide the large-time behavior of solutions showing the fluid density and the velocity converge to its averages exponentially fast as time goes to infinity.

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

“…Bouchut and James have developed a related but alternative theory of duality solutions [3,4] for solutions to (10). Their theory relies on properties of monotone solutions to (13), and they prove uniqueness under an assumption similar to (but stronger than) (12). The framework of [7] has been successfully applied to the 1D Euler-Poisson equations [6,48,49].…”

“…The major challenge is to appropriately treat the nonlocality from the alignment interactions. Unfortunately, the system cannot be connected to a scalar conservation law of the type (13), except for special cases, e.g. constant φ.…”

“…We then construct a unique weak solution to the 1D Euler-alignment system (14), using the entropy conditions for (15) as our selection principle. The intrinsic nonlocality embedded in (15) requires a significant advancement of the analytical techniques used for (13); this is true of the global wellposedness theory but is more significant in terms of the precise connection with (14). One distinctive feature of ( 15) is that ∂ x (A(M))…”

“…An alternative hydrodynamic limit can be obtained [37,38,39] by taking an isothermal ansatz f (x, v, t) = (2π) −d/2 ρ(x, t) exp(−|v − u(x, t)| 2 /2), in which case ∇ x • R becomes a pressure term ∇ x ρ. The resulting isothermal Euler-alignment system was investigated in [8,13]. A more general class of isentropic Euler-alignment system with a pressure term ∇ x (ρ γ ), γ ≥ 1 was considered in [14,12].…”

Pressureless Euler alignment system with control

“…For that reason, it is natural to take into account the measure-valued solutions for the global regularity. However, our main system (1.1) includes a nonlocal dissipation and this allows us to have global-in-time strong solutions under smallness assumptions on the initial data [31], see also [20] for the isothermal Euler alignment system. It is also even obtained a sharp critical threshold for the system (1.1) without control, i.e., φ ≡ 0, that leading to global regularity or finite-time blow-up of strong solutions [16] in one dimension.…”