On a weak solution for a transonic flow problem. (English) Zbl 0615.76070

The objective of this paper is to establish the existence of weak solutions for steady state two-dimensional inviscid irrotational compressible flow in an infinitely long channel or in the exterior of an airfoil given the speed upstream at \(\infty\). If the speed upstream is sufficiently small there exists a strong smooth flow because the governing equations are elliptic. This means that the speed of sound always exceeds the fluid speed. But beyond a certain speed at infinity the flow ceases to be purely elliptic and shocks may and do arise. This paper shows how to establish a weak existence theorem for such a flow by the addition of artificial viscosity v. The existence of a certain viscous flow with free boundary is presumed. The speed of flow is then assumed to be bounded away from cavitation speed and stagnation and the angle of flow is assumed bounded independent of the viscosity. Then weak existence of the flow solution in the limit \(v\to 0\) is proved.


76H05 Transonic flows
35L67 Shocks and singularities for hyperbolic equations
76N20 Boundary-layer theory for compressible fluids and gas dynamics
76L05 Shock waves and blast waves in fluid mechanics
35B40 Asymptotic behavior of solutions to PDEs
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