On the existence and stability of a global subsonic flow in a 3D infinitely long cylindrical nozzle

Abstract

This paper is concerned with the problem on the global existence and stability of a subsonic flow in an infinitely long cylindrical nozzle for the 3D steady potential flow equation. Such a problem was indicated by Courant-Friedrichs in [8, p. 377]: A flow through a duct should be considered as a steady, isentropic, irrotational flow with cylindrical symmetry and should be determined by solving the 3D potential flow equations with appropriate boundary conditions. By introducing some suitably weighted Hölder spaces and establishing a priori estimates, the authors prove the global existence and stability of a subsonic potential flow in a 3D nozzle when the state of subsonic flow at negative infinity is given.

Keywords

Subsonic flow / Potential flow equation / Bessel function / Weighted Hölder space / Global existence

References

[1] Amrouche C., Bonzon F.. Exterior problems in the half-space for the Laplace operator in weighted Sobolev spaces. J. Diff. Eqs., 2009, 246(5): 1894-1920

[2] Bers L.. Existence and uniqueness of a subsonic flow past a given profile. Comm. Pure Appl. Math., 1954, 7: 441-504

[3] Bers L.. Mathematical aspects of subsonic and transonic gas dynamics, 1958, New York, Chapman & Hall, London: John Wiley & Sons

[4] Bojarski B.. Subsonic flow of compressible fluid. Arch. Mech. Stos., 1966, 18: 497-520

[5] Chen G. Q., Dafermos C. M., Slemrod M. On two-dimensional sonic-subsonic flow. Comm. Math. Phys., 2007, 271(3): 635-647

[6] Ciavaldini J. F., Pogu M., Tournemine G.. Existence and regularity of stream functions for subsonic flows past profiles with a sharp trailing edge. Arch. Rational Mech. Anal., 1986, 93(1): 1-14

[7] Cook L. P., Newman E., Rimbey S. Sonic and subsonic axisymmetric nozzle flows. SIAM J. Appl. Math., 1999, 59(5): 1812-1824

[8] Courrant R., Friedrichs K. O.. Supersonic Flow and Shock Waves, 1948, New York: Interscience Publishers Inc.

[9] Dong G. C., Ou B.. Subsonic flows around a body in space. Comm. Part. Diff. Eqs., 1993, 18(1–2): 355-379

[10] Finn R., Gilbarg D.. Asymptotic behavior and uniqueness of plane subsonic flows. Comm. Pure Appl. Math., 1957, 10: 23-63

[11] Finn R., Gilbarg D.. Three-dimensional subsonic flows and asymptotic estimates for elliptic partial differential equations. Acta Math., 1957, 98: 265-296

[12] Gilbarg D., Tudinger N. S.. Elliptic Partial Differential Equations of Second Order, 1998 2nd ed. Berlin, New York: Springer-Verlag

[13] Grisvard P.. Elliptic problems in nonsmooth domains, 1985, Boston: Pitman (Advanced Publishing Program)

[14] Khairullina O. B.. Modelling subsonic vortex gas flows in channels of complex geometries. Russian J. Numer. Anal. Math. Modelling, 1998, 13(3): 191-218

[15] Liu L.. On subsonic compressible flows in a two-dimensional duct. Nonlinear Anal., 2008, 69(2): 544-556

[16] Morawetz C. S.. On the nonexistence of continuous transonic flows past profiles. Comm. Pure Appl. Math. I, 1956, 9: 45-68

[17] Rusak Z.. Subsonic flow around the leading edge of a thin aerofoil with a parabolic nose. European J. Appl. Math., 1994, 5: 283-311

[18] Shiffman M.. On the existence of subsonic flows of a compressible fluid. J. Rational Mech. Anal., 1952, 1: 605-652

[19] Watson G. N.. A Treatise on the Theory of Bessel Functions, 1944 2nd ed. Cambridge: Cambridge University Press

[20] Woods L. C.. Compressible subsonic flow in two-dimensional channels I, Basic mathematical theory. Aero. Quart., 1955, 6: 205-220

[21] Woods L. C.. Compressible subsonic flow in two-dimensional channels II, The application of the theory to problems of channel flow. Aero. Quart., 1955, 6: 254-276

[22] Woods L. C.. Compressible subsonic flow in two-dimensional channels with mixed boundary conditions. Quart. J. Mech. Appl. Math., 1954, 7: 263-282

[23] Xie C. J., Xin Z. P.. Global subsonic and subsonic-sonic flows through infinitely long nozzles. Indiana Univ. Math. J., 2007, 56(6): 2991-3023

[24] Xu G., Yin H. C.. Global transonic conic shock wave for the symmetrically perturbed supersonic flow past a cone. J. Diff. Eqs., 2008, 245: 3389-3432

[25] Xu G., Yin H. C.. Global multidimensional transonic conic shock wave for the perturbed supersonic flow past a cone. SIAM J. Math. Anal., 2009, 41(1): 178-218

[26] Xu, G. and Yin, H. C., Global subsonic flow in a 3D infinitely long curved nozzle, IMS at Nanjing University, 2008, preprint.

[27] Zhang P.. Remark on the regularities of Kato’s solutions to Navier-Storkes equations with initial data in L ^d(ℝ ^d). Chin. Ann. Math., 2008, 29B(3): 369-384