Long time behavior of solutions to the 3D compressible Euler equations with damping

The effect of damping on the large-time behavior of solutions to the Cauchy problem for the three-dimensional compressible Euler equations is studied. It is proved that damping prevents the development of singularities in small amplitude classical solutions, using an equivalent reformulation of the Cauchy problem to obtain effective energy estimates. The full solution relaxes in the maximum norm to the constant background state at a rate of t(-3/2). While the fluid vorticity decays to zero exponentially fast in time, the full solution does not decay exponentially. Formation of singularities is also exhibited for large data.