On the nonexistence of a global nontrivial subsonic solution in a 3D unbounded angular domain. (English) Zbl 1213.35306
The authors consider the steady isentropic irrotational flow with low Mach number described by the potential equation. The domain is unbounded and is one part of a 3D ramp. They focus on the non-existence of a global nontrivial subsonic solution to such equation. Under some assumptions about the flow they prove that there is no such solution to the system.
Reviewer: Ilya A. Chernov (Petrozavodsk)
MSC:
| 35L70 | Second-order nonlinear hyperbolic equations |
| 35L65 | Hyperbolic conservation laws |
| 35L67 | Shocks and singularities for hyperbolic equations |
| 76N15 | Gas dynamics (general theory) |
Keywords:
subsonic flow; potential equation; modified Bessel functions; weighted Hölder space; steady irrotational flowReferences:
| [1] | Abramowitz M, Stegun I A. Handbook of Mathematical Functions. New York: Dover Publications, Inc., 1972 · Zbl 0543.33001 |
| [2] | Amrouche C, Bonzon F. Exterior problems in the half-space for the Laplace operator in weighted Sobolev spaces. J Differential Equations, 2009, 246: 1723–2150 · Zbl 1169.35011 · doi:10.1016/j.jde.2008.11.021 |
| [3] | Azzam A. Smoothness properties of mixed boundary value problems for elliptic equations in sectionally smooth n-dimensional domains. Ann Polon Math, 1981, 40: 81–93 · Zbl 0485.35013 |
| [4] | Bers L. Existence and uniqueness of a subsonic flow past a given profile. Comm Pure Appl Math, 1954, 7: 441–504 · Zbl 0058.40601 · doi:10.1002/cpa.3160070303 |
| [5] | Bojarski B. Subsonic flow of compressible fluid. Arch Mech Stos, 1966, 18: 497–520 · Zbl 0141.42902 |
| [6] | Chen G Q, Zhang Y Q, Zhu D W. Existence and stability of supersonic Euler flows past Lipschitz wedges. Arch Ration Mech Anal, 2006, 181: 261–310 · Zbl 1121.76055 · doi:10.1007/s00205-005-0412-3 |
| [7] | Chen S X. Existence of local solution to supersonic flow past a three-dimensional wing. Adv Appl Math, 1992, 13: 273–304 · Zbl 0760.76040 · doi:10.1016/0196-8858(92)90013-M |
| [8] | Chen S X. Stability of oblique shock fronts. Sci China Ser A, 2002, 45: 1012–1019 · Zbl 1100.35506 · doi:10.1007/BF02879984 |
| [9] | Chen S X, Xin Z P, Yin H C. Global shock wave for the supersonic flow past a perturbed cone. Commun Math Phys, 2002, 228: 47–84 · Zbl 1006.76080 · doi:10.1007/s002200200652 |
| [10] | Courrant R, Friedrichs K O. Supersonic Flow and Shock Waves. New York: Interscience Publishers Inc., 1948 · Zbl 0041.11302 |
| [11] | Cui D C, Yin H C. Global conic shock wave for the steady supersonic flow past a cone: Isothermal case. Pacific J Math, 2007, 233: 257–289 · Zbl 1221.76175 · doi:10.2140/pjm.2007.233.257 |
| [12] | Cui D C, Yin H C. Global conic shock wave for the steady supersonic flow past a cone: Polytropic case. J Differential Equations, 2009, 246: 641–669 · Zbl 1171.35075 · doi:10.1016/j.jde.2008.07.031 |
| [13] | Dong G C, Ou B. Subsonic flows around a body in space. Comm Partial Differential Equations, 1993, 18: 355–379 · Zbl 0813.35013 · doi:10.1080/03605309308820933 |
| [14] | Finn R, Gilbarg D. Asymptotic behavior and uniqueness of plane subsonic flows. Comm Pure Appl Math, 1957, 10: 23–63 · Zbl 0077.18801 · doi:10.1002/cpa.3160100102 |
| [15] | Finn R, Gilbarg D. Three-dimensional subsonic flows and asymptotic estimates for elliptic partial differential equations. Acta Math, 1957, 98: 265–296 · Zbl 0078.40001 · doi:10.1007/BF02404476 |
| [16] | Gilbarg D, Tudinger N S. Elliptic Partial Differential Equations of Second Order, 2nd ed. Grundlehren der Mathematischen Wissenschaften, 224. Berlin-New York: Springer, 1998 |
| [17] | Grisvard P. Elliptic Problems in Nonsmooth Domains. Monographs and Studies in Mathematics, 24. Boston, MA: Pitman (Advanced Publishing Program), 1985 · Zbl 0695.35060 |
| [18] | Li T. On a free boundary problem. Chin Ann Math, 1980, 1: 351–358 · Zbl 0479.35075 |
| [19] | Lieberman G M. Mixed boundary value problems for elliptic and parabolic differential equation of second order. J Math Anal Appl, 1986, 113: 422–440 · Zbl 0609.35021 · doi:10.1016/0022-247X(86)90314-8 |
| [20] | Lien W C, Liu T P. Nonlinear stability of a self-similar 3-dimensional gas flow. Comm Math Phys, 1999, 204: 525–549 · Zbl 0945.76033 · doi:10.1007/s002200050656 |
| [21] | Rusak Z. Subsonic flow around the leading edge of a thin aerofoil with a parabolic nose. European J Appl Math, 1994, 5: 283–311 · Zbl 0826.76039 · doi:10.1017/S0956792500001479 |
| [22] | Schaeffer D G. Supersonic flow past a nearly straight wedge. Duke Math J, 1976, 43: 637–670 · Zbl 0356.76046 · doi:10.1215/S0012-7094-76-04351-9 |
| [23] | Shiffman M. On the existence of subsonic flows of a compressible fluid. J Rational Mech Anal, 1952, 1: 605–652 · Zbl 0048.19301 |
| [24] | Wang Z X, Guo D R. Introduction to Special Function. Beijing: Peking University Press, 2000 |
| [25] | Xin Z P, Yin H C. 3-Dimensional transonic shock in a nozzle. Pacific J Math, 2008, 236: 139–193 · Zbl 1232.35097 · doi:10.2140/pjm.2008.236.139 |
| [26] | Xu G, Yin H C. Global transonic conic shock wave for the symmetrically perturbed supersonic flow past a cone. J Differential Equations, 2008, 245: 3389–3432 · Zbl 1155.35062 · doi:10.1016/j.jde.2007.12.011 |
| [27] | 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: 178–218 · Zbl 1194.35261 · doi:10.1137/080712933 |
| [28] | Yin H C. Global existence of a shock for the supersonic flow past a curved wedge. Acta Math Sin (Engl Ser), 2006, 22: 1425–1432 · Zbl 1103.35075 · doi:10.1007/s10114-005-0611-8 |
| [29] | Zigangareeva L M, Kiselev O M. Structure of the long-range fields of plane symmetrical subsonic flows. Fluid Dynam, 2000, 35: 421–431 · Zbl 0979.76040 · doi:10.1007/BF02697756 |
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