On global smooth solutions of the 3D spherically symmetric Euler equations with time-dependent damping and physical vacuum

Abstract

In this paper, we consider the global existence and convergence of smooth solutions for the three dimensional spherically symmetric compressible Euler equations with time-dependent damping and physical vacuum. The damping coefficient decays with time and the sound speed is C 1/2-Hölder continuous across the physical vacuum boundary. Both the degeneration of the damping coefficient at time infinity and the non C 1 continuity of the sound speed across the vacuum boundary will cause difficulty in proving the global existence of smooth solutions. Under suitable assumptions on the decayed damping coefficients, the globally in-time smooth solutions and convergence to the modified Barenblatt solution will be given. Also obtained are the pointwise convergence rate of the density, velocity and the expanding rate of the physical vacuum boundary. Our result extends that in Zeng (2017 Arch. Ration. Mech. Anal. 226 33–82) by considering the degenerate damping coefficient instead of the constant damping coefficient and that in Pan (2021 Calc. Var. Partial Differ. Equ. 60 5) from the one dimensional case to the three dimensional case with spherically symmetric data.