Convergence rates to the Barenblatt solutions for the compressible Euler equations with time-dependent damping

Abstract

In this paper, we are concerned with the asymptotic behavior of L∞ weak entropy solutions for the compressible Euler equations with time-dependent damping and vacuum for any large initial data. This model describes the motion for the compressible fluid through a porous medium, and the friction force is time-dependent. We obtain that the density converges to the Barenblatt solution of a well-known porous medium equation with the same finite initial mass in L1 decay rate when 1+52<γ≤2,0≤λ<γ2−γ−1γ2+γ−1 or γ≥2,0≤λ<12γ+1 which partially improves and extends the previous work [14,6]. The proof is mainly based on the detailed analysis of the relative weak entropy, time-weighted energy estimates and the iterative method.