Uniqueness of symmetric steady subsonic flows in infinitely long divergent nozzles

Abstract

We prove that the spherically symmetric subsonic flows in an infinitely long straight divergent nozzle with arbitrary smooth cross-section are unique for the three-dimensional steady potential flow equation. The proof depends on an extreme principle for elliptic equations in an unbounded conical domain, under the assumption that the gradient of the solution is of order O 1 |x| as |x| → ∞. Similar result holds for steady subsonic Euler flows in two-dimensional infinitely long straight divergent nozzles. (2000). Primary 35J25 · 35B50; Secondary 76N10 · 76G25. Mathematics Subject Classification

Stability of spherically symmetric subsonic flows and transonic shocks under multidimensional perturbations

“…By the assumption that ∂ r ϕ ≥ 0 and the fact that ∂ r ϕ s > 0, we see l · (r 0 , 0, 0)/|r 0 | = ∂ r ϕ s + ∂ r ϕ > 0, hence on M 0 we have an oblique derivative condition of w. Remark 3.2. The uniqueness of symmetric subsonic solutions to potential flows in infinite conical nozzles had been proved in [22] by applying Harnack inequalities, see also [20] for the uniqueness result of the case that M = T 2 . The existence of isentropic irrotational subsonic flows in general three-dimensional largely-open nozzles was proved in [23], while the same existence problem for the three-dimensional Euler system still remains open.…”

Global Uniqueness of Transonic Shocks in Two-Dimensional Steady Compressible Euler Flows

“…Chen and H. Yuan [7] proved the transonic shock in a two-dimensional or three-dimensional straight duct is unique modulo a translation. L. Liu showed uniqueness of subsonic potential flow in various domains [22], see also [23]. By Proposition 3.1 in [9], in two-dimensional case, many of the uniqueness results on subsonic flows also hold for the Euler system, since under appropriate boundary conditions, the flow is in itself irrotational, i.e., governed by the potential flow equation.…”