In this paper, the smooth solution of the physical vacuum problem for the one dimensional compressible Euler equations with time-dependent damping is considered. Near the vacuum boundary, the sound speed is \(C^{1/2}\)-Hölder continuous. The coefficient of the damping depends on time, given by this form \(\frac{\mu }{(1+t)^\lambda }\), \(\lambda ,\ \mu >0\), which decays by order \(-\lambda \) in time. Under the assumption that \(0<\lambda<1,\ 0<\mu \) or \(\lambda =1,\ 2<\mu \), we will prove the global existence of smooth solutions and convergence to the modified Barenblatt solution of the related porous media equation with time-dependent dissipation and the same total mass when the initial data of the Euler equations is a small perturbation of that of the Barenblatt solution. The pointwise convergence rates of the density, velocity and the expanding rate of the physical vacuum boundary are also given. The proof is based on space-time weighted energy estimates, elliptic estimates and Hardy inequality in the Lagrangian coordinates. Our result is an extension of that in Luo–Zeng (Commun Pure Appl Math 69(7):1354–1396, 2016), where the authors considered the physical vacuum free boundary problem of the compressible Euler equations with constant-coefficient damping.

本文考虑一维可压缩的具有时变阻尼的欧拉方程的物理真空问题的光滑解。在真空边界附近，声速为\（C ^ {1/2} \）- Hölder连续。阻尼系数取决于时间，其形式为\（\ frac {\ mu} {（1 + t）^ \ lambda} \），\（\ lambda，\ \ mu> 0 \），衰减时间为及时订购\（-\ lambda \）。假设\（0 <\ lambda <1，\ 0 <\ mu \）或\（\ lambda = 1，\ 2 <\ mu \），当Euler方程的初始数据对Barenblatt的扰动很小时，我们将证明光滑解的整体存在性和相关多孔介质方程的修正Barenblatt解的收敛性，该扰动具有时变耗散且总质量相同解。还给出了物理真空边界的密度，速度和扩展率的逐点收敛速度。该证明是基于时空加权能量估计，椭圆估计和拉格朗日坐标中的Hardy不等式。我们的结果是对Luo-Zeng（Commun Pure Appl Math 69（7）：1354–139​​6，2016）中结果的扩展，作者在其中考虑了具有恒定系数阻尼的可压缩Euler方程的物理真空自由边界问题。