Well-posedness for physical vacuum free boundary problem of compressible Euler equations with time-dependent damping

Abstract In this article, we consider the well-posedness of the local smooth solutions to the physical vacuum free boundary problem of the cylindrical symmetric Euler equations with time-dependent damping − μ (1 + t) λ ρ u - { (1+t) ^ } u, μ > 0 0, and λ > 0 0. In this case, the free boundary is moving in the radial direction and the radial velocity will affect the angular velocity, but the axial velocity can be explicitly expressed by ω = ω 0 (1 + t) − μ = ₀ (1+t) ^- when λ = 1 =1 and ω = ω 0 e μ 1 − λ 1 − (1 + t) 1 − λ = ₀e^ {1- 1- (1+t) ^1- } when λ ≠ 1 1. Thus, when 0 λ 1 0 1, ω converges to 0 as t → + ∞ t +, and when λ > 1 1, ω converges to ω 0 e μ 1 − λ ₀e^ {1- } as t → + ∞ t +. This is essentially different from the case of μ = 1 =1, λ = 0 =0 in (Meng-Mai-Mei, JDE, 2022), whose axial velocity is only expressed by ω = ω 0 e − t = ₀e^-t. Moreover, at the moving boundary, the compressible Euler equations with time-dependent damping become a degenerate hyperbolic system. Thanks to the Hardy-type inequality and cut-off functions, we choose a suitable weighted Sobolev spaces to construct a priori estimates to overcome the degeneracy of the system in the Lagrangian coordinates, which confirms the existence of local smooth solution. Gronwall’s inequality guarantees the uniqueness of local smooth solution.