Global existence and convergence rates of solutions for the compressible Euler equations with damping

Abstract

The Cauchy problem for the 3D compressible Euler equations with damping is considered. Existence of global-in-time smooth solutions is established under the condition that the initial data is small perturbations of some given constant state in the framework of Sobolev space $ H^3(\mathbb{R}^{3}) $ only, but we don't need the bound of $ L^1 $ norm. Moreover, the optimal $ L^{2} $-$ L^{2} $ convergence rates are also obtained for the solution. Our proof is based on the benefit of the low frequency and high frequency decomposition, here, we just need spectral analysis of the low frequency part of the Green function to the linearized system, so that we succeed to avoid some complicate analysis.