Global relaxation to the fractional porous medium equation from the pressureless Euler–Riesz system

In this paper, we justify the validity of global-in-time convergence rigorously from the pressureless relaxed Euler-Riesz system to the frictional porous medium model in Rd (d ≥ 1). For initial data close to a constant equilibrium in critical Besov spaces, we prove the global existence and uniqueness of strong solutions for the pressureless damped Euler-Riesz system, uniformly with respect to the relaxation parameter. Furthermore, we derive global-in-time error estimates between solutions of the pressureless Euler-Riesz system and the porous medium model for ill-prepared data, with an explicit convergence rate. The proof relies on the non-local hypocoercivity argument, which enables us to track the regularity evolution with respect to the relaxation parameter and exhibit the frictional dissipative mechanism arising from Riesz interactions.