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In this paper, the smooth solution of the physical vacuum problem for the onedimensional compressible Euler equations with time-dependent damping isconsidered. Near the vacuum boundary, the sound speed is $C^{1/2}$-H\"{o}ldercontinuous. 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 solutionsand convergence to the modified Barenblatt solution of the related porous mediaequation with time-dependent dissipation and the same total mass when theinitial data of the Euler equations is a small perturbation of that of theBarenblatt solution. The pointwise convergence rates of the density, velocityand the expanding rate of the physical vacuum boundary are also given. Theproof is based on space-time weighted energy estimates, elliptic estimates andHardy inequality in the Lagrangian coordinates. Our result is an extension ofthat in Luo-Zeng [Comm. Pure Appl. Math. 69 (2016), no. 7, 1354-1396], wherethe authors considered the physical vacuum free boundary problem of thecompressible Euler equations with constant-coefficient damping.

calculus of variations and partial differential equations

Global existence and convergence to the modified Barenblatt solution for  the compressible Euler equations with physical vacuum and time-dependent  damping