Stability and asymptotic behavior of transonic flows past wedges for the full Euler equations

Abstract

The existence, uniqueness, and asymptotic behavior of steady transonic flows past a curved wedge, involving transonic shocks, governed by the two-dimensional full Euler equations are established. The stability of both weak and strong transonic shocks under the perturbation of the upstream supersonic flow and the wedge boundary is proved. The problem is formulated as a one-phase free boundary problem, in which the transonic shock is treated as a free boundary. The full Euler equations are decomposed into two algebraic equations and a first-order elliptic system of two equations in Lagrangian coordinates. With careful elliptic estimates by using appropriate weighted Hölder norms, the iteration map is defined and analyzed, and the existence of its fixed point is established by performing the Schauder fixed point argument. The careful analysis of the asymptotic behavior of the solutions reveals particular characters of the full Euler equations.

Supersonic flow onto solid wedges, multidimensional shock waves and free boundary problems

“…However, working on the potential function requires at least the Lipschitz estimate of the potential function to keep the subsonicity of the flow. In our recent paper [7], we have directly employed the decomposition of the full Euler equations into two algebraic equations and a first-order elliptic system of two equations and have established the stability and asymptotic behavior of transonic flows for Problem 2.2 (WT)-(ST) in a weighted Hölder space.…”

“…To state the results, following [7], we need to introduce the weighed Hölder norms in the subsonic domain Ω, where Ω is either a truncated triangular domain or an unbounded domain with the vertex at origin O and one side as the wedge boundary. There are two weights: One is the distance function to origin O and the other is to the wedge boundary ∂W.…”

“…In Chen-Chen-Feldman [24], the weaker transonic shock, which corresponds to arc Ň T S, was first investigated by Approach I. Then, in [25], the weak and strong transonic shocks, which correspond to arcs Ň T S and Ŋ T H , respectively, were solved, by Approach II which is described in §3.3 below, so that the existence, uniqueness, stability, and asymptotic behavior of subsonic solutions of Problem 3.2 (WT) & (ST) in a weighted Hölder space were obtained.…”

“…Moreover, ν 0 w is the interior (for Ω 0 ) unit normal to BW 0 , and τ 0 w is the tangential unit vector to BW 0 , where BW 0 and Ω 0 are defined by (3.13) and (3.17) for the background solution pU 0 , U 0 q, i.e., u ¨τ 0 w and u ¨ν0 w are components u 1 and u 2 of u in the coordinates rotated clockwise by angle θ w , so that the background downstream flow becomes horizontal. Theorem 3.1 (Chen-Chen-Feldman [25]). Let pU 0 , U 0 q be a constant transonic solution for the wedge angle θ w P p0, θ d w q.…”

“…More specifically, we establish the following theorem in the rotated coordinates: Theorem 3.2 (Chen-Chen-Feldman [25]).…”