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This article addresses a fundamental concern regarding the incompressibleapproximation of fluid motions, one of the most widely used approximations influid mechanics. Common belief is that its accuracy is $O(\epsilon)$ where$\epsilon$ denotes the Mach number. In this article, however, we prove an$O(\epsilon^2)$ accuracy for the incompressible approximation of theisentropic, compressible Euler equations thanks to several decouplingproperties. At the initial time, the velocity field and its first timederivative are of $O(1)$ size, but the boundary conditions can be as stringentas the solid-wall type. The fast acoustic waves are still $O(\epsilon)$ inmagnitude, since the $O(\epsilon^2)$ error is measured in the sense of Lerayprojection and more physically, in time-averages. We also show when a passivescalar is transported by the flow, it is $O(\epsilon^2)$ accurate {\itpointwise in time} to use incompressible approximation for the velocity fieldin the transport equation.

Improved Accuracy of Incompressible Approximation of Compressible Euler Equations