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

• Published in: Journal of mathematical fluid mechanics Vol. 27; no. 4
• Main Authors: Chi, Meiling, Shou, Ling-Yun, Xu, Jiang
• Format: Journal Article
• Language: English
• Published: Heidelberg Springer Nature B.V 01.11.2025
• Subjects: Cauchy problems; Compressibility; Damping; Darcys law; Density; Eigenvalues; Euler-Lagrange equation; Thermodynamics
• ISSN: 1422-6928; 1422-6952
• DOI: 10.1007/s00021-025-00964-w

We are concerned with a system governing the evolution of the pressureless compressible Euler equations with Riesz interaction and damping in Rd (d≥1), where the interaction force is given by ∇(-Δ)(α-d)/2ρ with d-2<α<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 Lp framework. Precisely, the density and its σ-order derivative converge to the equilibrium at the Lp-rate (1+t)-(σ-σ1)/(α-d+2) with -d/p-1≤σ1<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+∇Λα-dρ associated with the Darcy law are introduced to overcome the difficulty from the absence of hyperbolic symmetrization for first-order dissipative systems.