Stability of smooth solutions for the compressible Euler equations with time-dependent damping and one-side physical vacuum

In this paper, the one-side physical vacuum problem for the one dimensional compressible Euler equations with time-dependent damping is considered. Near the physical vacuum boundary, the sound speed is C 1 / 2 -Holder continuous. The coefficient of the time-dependent damping is given by μ ( 1 + t ) λ , ( 0 λ , 0 μ ) which decays by order −λ in time. First we give an one-side physical vacuum background solution whose density and velocity have a growing order with respect to time. Then the main purpose of this paper is to prove the stability of this background solution under the assumption that 0 λ 1 , 0 μ or λ = 1 , 2 μ . The pointwise convergence rate of the density, velocity and the expanding rate of the physical vacuum boundary are also given. The proof is based on the space-time weighted energy estimates, elliptic estimates and the Hardy inequality in the Lagrangian coordinates.