Improved Accuracy of Incompressible Approximation of Compressible Euler Equations

This article addresses a fundamental concern regarding the incompressible approximation of fluid motions, one of the most widely used approximations in fluid mechanics. Common belief is that its accuracy is $O(\varepsilon)$ where $\varepsilon$ denotes the Mach number. In this article, however, we prove an $O(\varepsilon^2)$ accuracy for the incompressible approximation of the isentropic, compressible Euler equations thanks to several decoupling properties. At the initial time, the velocity field and its first time derivative are of $O(1)$ size, but the boundary conditions can be as stringent as the solid-wall type. The fast acoustic waves are still $O(\varepsilon)$ in magnitude, since the $O(\varepsilon^2)$ error is measured in the sense of Leray projection and more physically, in time-averages. We also show when a passive scalar is transported by the flow, it is $O(\varepsilon^2)$ accurate point-wise in time to use incompressible approximation for the velocity field in the transport equation.