Abstract

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

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