The incompressible limit and the initial layer of the compressible Euler equation in ℝⁿ+

We consider the incompressible limit of the compressible Euler equation in the half-space ℝⁿ₊. It is proved that the solutions of the non-dimensionalized compressible Euler equation converge to the solution of the incompressible Euler equation when the Mach number tends to zero. If the initial data v∞₀ do not satisfy the condition '∇·v∞₀ = 0', then the initial layer will appear.