On Multi-Dimensional Sonic-Subsonic Flow

TL;DR

This paper develops a new mathematical framework to analyze steady irrotational flows with sonic and subsonic speeds in multiple dimensions, proving the existence of solutions in complex flow scenarios.

Contribution

It introduces a compensated compactness framework for n-dimensional sonic-subsonic flows, establishing the first results on the sonic-subsonic limit for dimensions three and higher.

Findings

Existence of sonic-subsonic weak solutions for n-dimensional flows.

Framework applicable to flows past obstacles and through nozzles.

First known results on sonic-subsonic limits in three or more dimensions.

Abstract

In this paper, a compensated compactness framework is established for sonic-subsonic approximate solutions to the $n$-dimensional$(n\geq 2)$ Euler equations for steady irrotational flow that may contain stagnation points. This compactness framework holds provided that the approximate solutions are uniformly bounded and satisfy $H^{-1}_{loc}(\Omega)$ compactness conditions. As illustration, we show the existence of sonic-subsonic weak solution to n-dimensional$(n\geq 2)$ Euler equations for steady irrotational flow past obstacles or through an infinitely long nozzle. This is the first result concerning the sonic-subsonic limit for $n$-dimension$(n\geq 3)$.