Abstract: Using the method of energy integral, we discussed the global existence and strong relaxation limit of smooth solutions for higher dimensional Euler equations with damping when the initial values were sufficiently small, and  obtained the uniform a priori estimates of the solutions. We proved that the asymptotic behavior of the global solution was controlled by the porous media equation when the relaxation time tended to zero.

Key words: Euler equation, relaxation, global smooth solution, nonlinear diffusion equation