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 , where denotes the Mach number. In this article, however, we prove an O 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 in magnitude, since the O 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 2 accurate pointwise in time to use incompressible approximation for the velocity field in the transport equation.