Compressible Euler equations with time-dependent damping in the critical regularity setting: global well-posedness and strong relaxation limit

We investigate the relaxation problem and the diffusion phenomenon for the compressible Euler system with a time-dependent damping coefficient of the form $\tfrac{\mu}{(1+t)^{\lambda}}$ in $\mathbb{R}^d$ $(d \geq 1)$. We establish uniform regularity estimates with respect to the relaxation parameter $\varepsilon$ and prove the global well-posedness of classical solutions to the Cauchy problem. In addition, we justify the global-in-time strong convergence of the solutions towards those of a general porous medium-type diffusion system, with an explicit rate of convergence, and for ill-prepared initial data. The core of our proof relies on a refined hypocoercivity framework combined with a new time-dependent frequency decomposition, both adapted to handle damping terms with time-dependent coefficients. This enables us to treat the overdamped regime $\lambda \in (-\infty,0)$ and the underdamped regime $\lambda \in (0,1)$ for any $\mu>0$, and also the borderline critical case $\lambda=1$ under the improved condition $\mu>2\varepsilon^2$.