The pressureless damped Euler-Riesz system in the critical regularity framework

Abstract: We are concerned with a system governing the evolution of the pressureless compressible Euler equations with Riesz interaction and damping in $\mathbb{R}^{d}$ ($d\geq1$), where the interaction force is given by $\nabla(-\Delta)^{\smash{\frac{\alpha-d}{2}}}(\rho-\bar{\rho})$ with $d-2<\alpha<d$. 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 $\dot{B}^{\sigma_1}_{p,\infty}$ with $-d/p-1\leq \sigma_1<d/p-1$, it is shown that the $L^p$-norm of the $\sigma$-order derivative for the density converges to its equilibrium at the rate $(1+t)^{-\smash{\frac{\sigma-\sigma_1}{\alpha-d+2}}}$, which coincides with that of the fractional heat kernel.