Two Dimensional Subsonic Euler Flow. Past a Wall or a Symmetric Body

TL;DR

This paper proves the existence and uniqueness of two-dimensional subsonic Euler flows past a wall or symmetric body under certain conditions, identifying critical densities and describing flow properties.

Contribution

It establishes the conditions for existence and uniqueness of subsonic flows past walls or symmetric bodies, including critical density thresholds and flow behavior details.

Findings

Existence of subsonic flows for densities above a critical value

Flows have large vorticity and positive horizontal velocity

Asymptotic behavior characterized by integral estimates

Abstract

The existence and uniqueness of two dimensional steady compressible Euler flows past a wall or a symmetric body are established. More precisely, given positive convex horizontal veloicty in the upstream, there exists a critical value $\rho_{cr}$ such that if the incoming density in the upstream is larger than $\rho_{cr}$, then there exists a subsonic flow past a wall. Furthermore, $\rho_{cr}$ is critical in the sense that there is no such subsonic flow if the density of the incoming flow is less than $\rho_{cr}$. The subsonic flows possess large vorticity and positive horizontal velocity above the wall except at the corner points on the boundary. Moreover, the existence and uniqueness of a two dimensional subsonic Euler flow past a symmetric body are also obtained when the incoming velocity field is a general small perturbation of a constant velocity field and the density of the incoming flow is larger than a critical value. The asymptotic behavior of the flows is obtained with the aid of some integral estimates for the velocity field and its far field states.