Low Mach Number Limit of Multidimensional Steady Flows on the Airfoil Problem

In this paper, we justify the low Mach number limit of the steady irrotational Euler flows for the airfoil problem, which is the first result for the low Mach number limit of the steady Euler flows in an exterior domain. The uniform estimates on the compressibility parameter $\varepsilon$, which is singular for the flows, are established via a variational approach based on the compressible-incompressible difference functions. The limit is on the H\"{o}lder space and is unique. Moreover, the convergence rate is of order $\varepsilon^2$. It is noticeable that, due to the feature of the airfoil problem, the extra force dominates the asymptotic decay rate of the compressible flow to the infinity. And the effect of extra force vanishes in the limiting process from compressible flows to the incompressible ones, as the Mach number goes to zero.