Incompressible Limit of Solutions of Multidimensional Steady Compressible Euler Equations

TL;DR

This paper develops a compactness framework to analyze the incompressible limit of steady compressible Euler equations in multiple dimensions, providing new existence results for incompressible flows in nozzles.

Contribution

It introduces a novel compactness approach based on weak estimates, clarifies the incompressibility derivation for different Euler flows, and establishes new limit theorems and existence results.

Findings

Established two new incompressible limit theorems for steady Euler flows.

Proved existence of solutions for multidimensional steady incompressible Euler equations.

Demonstrated the effectiveness of weak estimates in compactness arguments.

Abstract

A compactness framework is formulated for the incompressible limit of approximate solutions with weak uniform bounds with respect to the adiabatic exponent for the steady Euler equations for compressible fluids in any dimension. One of our main observations is that the compactness can be achieved by using only natural weak estimates for the mass conservation and the vorticity. Another observation is that the incompressibility of the limit for the homentropic Euler flow is directly from the continuity equation, while the incompresibility of the limit for the full Euler flow is from a combination of all the Euler equations. As direct applications of the compactness framework, we establish two incompressible limit theorems for multidimensional steady Euler flows through infinitely long nozzles, which lead to two new existence theorems for the corresponding problems for multidimensional steady incompressible Euler equations.