Multiple scales and singular limits of perfect fluids

TL;DR

This paper investigates the singular limits of a scaled rotating compressible Euler system, demonstrating convergence to the incompressible Euler system under specific conditions and extending the approach to Navier-Stokes systems.

Contribution

It provides a detailed analysis of the singular limits for a scaled barotropic Euler system with multiple parameters, including the convergence results and adaptation to Navier-Stokes equations.

Findings

Convergence to incompressible Euler system for well-prepared data.

Extension of techniques to Navier-Stokes system in subcritical range.

Identification of limit behavior in singular parameter regimes.

Abstract

In this article our goal is to study the singular limits for a scaled barotropic Euler system modelling a rotating, compressible and inviscid fluid, where Mach number $=\epsilon^m $, Rossby number $=\epsilon $ and Froude number $=\epsilon^n $ are proportional to a small parameter $\epsilon\rightarrow 0$. The fluid is confined to an infinite slab, the limit behaviour is identified as the incompressible Euler system. For \emph{well--prepared} initial data, the convergence is shown on the life span time interval of the strong solutions of the target system, whereas a class of generalized \emph{dissipative solutions} is considered for the primitive system. The technique can be adapted to the compressible Navier--Stokes system in the subcritical range of the adiabatic exponent $\gamma$ with $1<\gamma\leq\frac{3}{2}$, where the weak solutions are not known to exist.