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

TL;DR

This paper proves the global existence and uniqueness of strong solutions for the compressible Euler equations with nonlocal dissipation, derived from kinetic models, and analyzes their exponential convergence to equilibrium.

Contribution

It establishes the global well-posedness and large-time behavior of solutions for a nonlocal dissipative Euler system derived from kinetic models, using a perturbation approach.

Findings

Global existence and uniqueness of strong solutions

Exponential decay of solutions to equilibrium

Derivation from kinetic Cucker-Smale model

Abstract

This paper studies the global existence and uniqueness of strong solutions and its large-time 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.