The Pressureless Damped Euler-Riesz System in the Critical Regularity Framework

We are concerned with a system governing the evolution of the pressureless compressible Euler equations with Riesz interaction and damping in \(\mathbb {R}^{d}\) (\(d\ge 1\)), where the interaction force is given by \(\nabla (-\Delta )^{(\alpha -d)/2}\rho \) with \(d-2<\alpha <d\). It is observed by the eigenvalue analysis that the density exhibits fractional heat diffusion behavior at low frequencies, which enables us to establish the global existence and large-time behavior of solutions to the Cauchy problem in the critical \(L^p\) framework. Precisely, the density and its \(\sigma \)-order derivative converge to the equilibrium at the \(L^p\)-rate \((1+t)^{-(\sigma -\sigma _1)/(\alpha -d+2)}\) with \(-d/p-1\le \sigma _1< d/p-1\), consistent with the rate of solutions for the frictional heat equation. A non-local hypercoercivity argument and the effective unknown \(z=u+\nabla \Lambda ^{\alpha -d}\rho \) associated with the Darcy law are introduced to overcome the difficulty from the absence of hyperbolic symmetrization for first-order dissipative systems.