Half space problem for Euler equations with damping in 3-D

In this paper, we consider the half space problem for Euler equations with damping in 3-D. We restudy the fundamental solution for the Cauchy problem to obtain an exponentially sharp pointwise structure and a clear decomposition of the singular-regular components. Later, both Green's function for initial boundary value problem and fundamental solutions for Cauchy problems are investigated in the transformed domain after Laplace transform. The symbols are obtained and a connection between Green's function and fundamental solutions are established for the pointwise space-time structure of Green's function. Finally, the sharp estimates for Green's function together with a priori estimates from the energy method for high order derivatives result in the nonlinear stability of the solution and also the decaying rates.