Transonic Shocks in Compressible Flow Passing a Duct for ThreeDimensional Euler Systems
 Shuxing Chen,
 Hairong Yuan
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In this paper we study the transonic shock in steady compressible flow passing a duct. The flow is a given supersonic one at the entrance of the duct and becomes subsonic across a shock front, which passes through a given point on the wall of the duct. The flow is governed by the threedimensional steady full Euler system, which is purely hyperbolic ahead of the shock and is of elliptic–hyperbolic composed type behind the shock. The upstream flow is a uniform supersonic one with the addition of a threedimensional perturbation, while the pressure of the downstream flow at the exit of the duct is assigned apart from a constant difference. The problem of determining the transonic shock and the flow behind the shock is reduced to a freeboundary value problem. In order to solve the freeboundary problem of the elliptic–hyperbolic system one crucial point is to decompose the whole system to a canonical form, in which the elliptic part and the hyperbolic part are separated at the level of the principal part. Due to the complexity of the characteristic varieties for the threedimensional Euler system the calculus of symbols is employed to complete the decomposition. The new ingredient of our analysis also contains the process of determining the shock front governed by a pair of partial differential equations, which are coupled with the threedimensional Euler system.
 Cole J.D., Cook L.P. Transonic Aerodynamics, vol. 30. NorthHolland, Amsterdam, 1986
 Chen, G.Q., Feldman, M. (2003) Multidimensional transonic shocks and free boundary problems for nonlinear equations of mixed type. J. Am. Math. Soc. 16: pp. 461494 CrossRef
 Chen, G.Q., Feldman, M. (2004) Steady transonic shocks and free boundary problems in infinite cylinders for the Euler equations. Commun. Pure Appl. Math. 57: pp. 310356 CrossRef
 Čanić, S., Keyfitz, B.L., Lieberman, G.M. (2000) A proof of existence of perturbed steady transonic shocks via a free boundary problem. Commun. Pure Appl. Math. 53: pp. 484511 CrossRef
 Canic, S., Kerfitz, B., Kim, E.H. (2000) A free boundary problems for unsteady transonic small disturbance equation: transonic regular reflection. Methods Appl. Anal. 7: pp. 313336
 Canic, S., Kerfitz, B., Kim, E.H. (2002) A free boundary problems for a quasilinear degenerate elliptic equation: transonic regular reflection. Commun. Pure Appl. Math. 55: pp. 7192 CrossRef
 Chen, S. (1980) On the initialboundary value problem for quasilinear symmetric hyperbolic system and applications. Chin. Ann. Math. 1: pp. 511522
 Chen, S. (2002) Stability of oblique shock fronts. Sci. China 45: pp. 10121019 CrossRef
 Chen, S. (2005) Stability of transonic shock fronts in twodimensional Euler systems. Trans. Am. Math. Soc. 357: pp. 287308 CrossRef
 Courant, R., Friedrichs, K.O. (1948) Supersonic Flow and Shock Waves. Interscience, Publishers Inc., New York
 Glaz, H.M., Liu, T.P. (1984) The asymptotic analysis of wave interactions and numerical calculations of transonic flow. Adv. Appl. Math. 5: pp. 111146 CrossRef
 Gamba, I.M., Morawetz, C.S. (1996) A viscous approximations for a 2D steady semiconductor or transonic gas dynamic flow: existence theorem for potential flow. Commun. Pure Appl. Math. 49: pp. 9991049 CrossRef
 Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equation of Second Order, 2nd edn. Grundlehren der Mathematischen Wissenschaften, 224. Springer, Berlin, 1983
 Hörmander, L. (1985) The Analysis of Linear Partial Differential Operators, 3. Springer, Berlin
 Kuz’min, A.G. (2002) Boundary Value Problems for Transonic Flow. John Wiley, London
 Liu, T.P. (1982) Nonlinear stability and instability of transonic flows through a nozzle. Commun. Math. Phys. 83: pp. 243260 CrossRef
 Mcowen, R.C. (2003) Partial Differential Equations, Methods and Aplications. Pearson Education, Upper Saddle River, NJ
 Morawetz, C.S. (1956) On the nonexistence of continuous transonic flows past profiles. I, II, III. Commun. Pure Appl. Math. 9: pp. 4568 CrossRef
 Morawetz, C.S. (1957) On the nonexistence of continuous transonic flows past profiles. I, II, III. Commun. Pure Appl. Math. 10: pp. 107131 CrossRef
 Morawetz, C.S. (1958) On the nonexistence of continuous transonic flows past profiles. I, II, III. Commun. Pure Appl. Math. 11: pp. 129144 CrossRef
 Morawetz, C.S. (1964) Nonexistence of transonic flows past profiles. Commun. Pure Appl. Math. 17: pp. 357367 CrossRef
 Showalter, R.E. (1977) Hilbert Space Methods for Partial Differential Equations. Pitman, London–San Francisco–Melbourne
 Smoller, J. (1994) Shock Waves and Reaction–Diffusion Equations. Springer, New York
 Xin, Z.P., Yin, H.C. (2005) Transonic shock in a nozzle I: twodimensional case. Commun. Pure Appl. Math. 58: pp. 9991050 CrossRef
 Zeidlerm, E. (1986) Nonlinear Functional Analysis and its Applications. vol. 1. FixedPoint Theorems, Springer
 Title
 Transonic Shocks in Compressible Flow Passing a Duct for ThreeDimensional Euler Systems
 Journal

Archive for Rational Mechanics and Analysis
Volume 187, Issue 3 , pp 523556
 Cover Date
 20080301
 DOI
 10.1007/s002050070079z
 Print ISSN
 00039527
 Online ISSN
 14320673
 Publisher
 SpringerVerlag
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 Authors

 Shuxing Chen ^{(1)}
 Hairong Yuan ^{(1)}
 Author Affiliations

 1. School of Mathematical Sciences and Key Laboratory for Nonlinear Sciences (Fudan University, Ministry of Education), Fudan University, Shanghai, 200433, China