The pressureless damped Euler–Riesz equations

Abstract

We are concerned with a system governing the evolution of the pressureless compressible Euler equations with Riesz interaction and damping in R d (d ≥ 1), where the interaction force is given by ∇(−∆)Referring to the standard dissipative structure of first-order hyperbolic systems, the purpose of this paper is to investigate the weaker dissipation effect arising from the interaction force and to establish the global existence and large-time behavior of solutions to the Cauchy problem in the critical L p framework. More precisely, it is observed by the spectral analysis that the density behaves like fractional heat diffusion at low frequencies. Furthermore, if the low-frequency part of the initial perturbation is bounded in some Besov space Ḃσ1 p,∞ with −d/p − 1 ≤ σ 1 < d/p − 1, it is shown that the L p -norm of the σ-order derivative for the density converges to its equilibrium at the rate (1 + t) − σ−σ 1 α−d+2 , which coincides with that of the fractional heat kernel.