On the existence and stability of 2-D perturbed steady subsonic circulatory flows

Abstract

In this paper, under the generalized conservation condition of mass flux in a unbounded domain, we are concerned with the global existence and stability of a perturbed subsonic circulatory flow for the twodimensional steady Euler equation, which is assumed to be isentropic and irrotational. Such a problem can be reduced into a second order quasi-linear elliptic equation on the stream function in an exterior domain with a Dirichlet boundary value condition on the circular body and a stability condition at infinity. The key ingredient is establishing delicate weighted Hölder estimates to obtain the infinite behaviors of the flow under physical assumption.

The Global Stability of 2-D Subsonic Circulatory Flows for the Steady Isothermal Gas

“…As in (Cui & Li, 2011), through application of Green formula on the first equation in (1), the mass-flux of the circulatory flow should be invariant along each radial ray l which starts from the boundary ∂Ω, so the flow should satisfy the following generalized mass-flux condition…”

“…We can get the circulatory subsonic solution (ψ 0 (r), ρ 0 (r)) of the system (7) in the domain Ω 0 = {x : |x| > 1} by the analogous methods as in (Cui & Li, 2011), with each streamline being a circle and the center being at the origin just as illustrated in (Courant & Friedrichs, 1948). For the specific details, one can see the appendix in this paper.…”

“…As in (Cui & Li, 2011;Gilbarg & Tudinger, 1998), based on this estimate we can use the continuity method to solve the linearized problem of (15). Furthermore, by this estimate together with the standard fixed-point argument in (Gilbarg & Tudinger, 1998), we can arrive at the existence and uniqueness of the solution of the nonlinear problem (15).…”

“…This paper is a complement of our work in (Cui & Li, 2011) where we have established the global subsonic circulatory solution for the polytropic gas. In this paper, we are concerned with the global stability of the 2-D subsonic circulatory flow around a perturbed circular body for the isothermal gas.…”

“…L.Bers (see Bers, 1945) uses the pesudo-complex analysis method to obtain the global existence of the subsonic circulatory flow under the conditions that the adiabatic exponent γ = −1. In (Cui & Li, 2011), we have established the global existence of the subsonic circulatory flows solutions for inviscid gases with adiabatic exponent 1 < γ < 3. In the present paper, our goal is to establish the global existence and stability of subsonic circulatory flows solution for the subsonic isothermal gas around a perturbed circular body.…”