Incompressible and fast rotation limits for 3D compressible rotating Euler system with general initial data

Abstract This paper is concerned with the low Mach and Rossby number limits of 3D compressible rotating Euler equations with ill‐prepared initial data in the whole space. More precisely, the initial data is the sum of a 3D part and a 2D part. With the help of a suitable intermediate system, we perform this singular limit rigorously with the target system being a 2D QG‐type. This particularly gives an affirmative answer to the question raised by Ngo and Scrobogna. As a by‐product, our proof gives a rigorous justification of the convergence from the 2D inviscid rotating shallow water equations to the 2D QG equations in the whole space.