Existence of global smooth solutions for Euler equations with symmetry

Eventhough existence of global smooth solutions for one dimensional quasilinear hyperbolic systems has been well established, much less is known about the corresponding results for higher dimensional cases. In this paper, we study the existence of global smooth solutions for the initial-boundary value problem of Euler equtions satisfying gamma-law with damping and axisymmetry, or spherical symmetry. When the damping is strong enough, we give some sufficient conditions for existence of global smooth solutions as 1 < gamma < 5/3 and 5/3 < gamma < 3. The proof is based on technical estimation of the C-1 norm of the solutions.