The incompressible limit and the initial layer of the compressible Euler equation (Q1089214)

The author considers the Euler equations for isentropic compressible fluid flow in \(R^ n\), \(n\geq 3\), and discuss the limit of solutions as the Mach number \(\lambda\) tends to infinity. In a previous paper, \textit{S. Klainerman} and \textit{A. Majda} [Commun. Pure Appl. Math. 35, 629-651 (1982; Zbl 0478.76091)] considered this same problem; in particular they proved that unique solutions exist for all large \(\lambda\) on the time interval [0,T] independent of \(\lambda\), and that if the initial datum is incompressible, then the solutions converge as \(\lambda \to +\infty\) uniformly on [0,T] to a solution of the incompressible Euler equation. In the present paper the author shows that even if the initial datum is not incompressible, the limit still exists and satisfies the incompressible Euler equation. Due to the development of initial layer, the convergence is not uniform near \(t=0\).