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Morawetz, Cathleen Synge

Mixed equations and transonic flow. (English) Zbl 1055.35093

J. Hyperbolic Differ. Equ. 1, No. 1, 1-26 (2004).
Summary: This paper reviews the present situation with existence and uniqueness theorems for mixed equations and their application to the problems of transonic flow. Some new problems are introduced and discussed.
After a very brief discussion of time-dependent flows the steady state and its history is described. Early work on mixed equations and their connection to 2D flow, up the problem of shocks, the construction of good airfoils and the relevant boundary value problems are described. Then, we look at what two linear perturbation problems could tell us about the flow. We describe other examples of fluid problems giving rise to similar problems. Moreover, the uniqueness by a conservation law and the existence proofs by Friedrichs’ multipliers are considered. A proof is given of the existence of a steady flow corresponding to some of the previous examples but the equations have been modified to a higher order system with a small parameter which when set to zero yields the equations of transonic flow. It remains to show that this formal limit really holds.
Much has been left out especially modern computational results and the text reflects the particular interests of the author.

Cited in 62 Documents

MSC:

35Q35 PDEs in connection with fluid mechanics
76H05 Transonic flows
35M10 PDEs of mixed type
76B10 Jets and cavities, cavitation, free-streamline theory, water-entry problems, airfoil and hydrofoil theory, sloshing

Keywords:

Mixed equations; Tricomi equation; Frankl problem; airfoil construction; self-similar time dependent flow; nonlinear Schrödinger equation; existence; uniqueness; relaxation model
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