Multidimensional transonic shocks and free boundary problems for nonlinear equations of mixed type

Chen, Gui-Qiang; Feldman, Mikhail

doi:10.1090/S0894-0347-03-00422-3

Multidimensional transonic shocks and free boundary problems for nonlinear equations of mixed type
HTML articles powered by AMS MathViewer

by Gui-Qiang Chen and Mikhail Feldman

J. Amer. Math. Soc. 16 (2003), 461-494

DOI: https://doi.org/10.1090/S0894-0347-03-00422-3

Published electronically: January 24, 2003

PDF | Request permission

Abstract:

We establish the existence and stability of multidimensional transonic shocks for the Euler equations for steady potential compressible fluids. The Euler equations, consisting of the conservation law of mass and the Bernoulli law for the velocity, can be written as a second-order, nonlinear equation of mixed elliptic-hyperbolic type for the velocity potential. The transonic shock problem can be formulated into the following free boundary problem: The free boundary is the location of the transonic shock which divides the two regions of smooth flow, and the equation is hyperbolic in the upstream region where the smooth perturbed flow is supersonic. We develop a nonlinear approach to deal with such a free boundary problem in order to solve the transonic shock problem. Our results indicate that there exists a unique solution of the free boundary problem such that the equation is always elliptic in the downstream region and the free boundary is smooth, provided that the hyperbolic phase is close to a uniform flow. We prove that the free boundary is stable under the steady perturbation of the hyperbolic phase. We also establish the existence and stability of multidimensional transonic shocks near spherical or circular transonic shocks.

References

H. W. Alt and L. A. Caffarelli, Existence and regularity for a minimum problem with free boundary, J. Reine Angew. Math. 325 (1981), 105–144. MR 618549
Hans Wilhelm Alt, Luis A. Caffarelli, and Avner Friedman, A free boundary problem for quasilinear elliptic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 11 (1984), no. 1, 1–44. MR 752578
Hans Wilhelm Alt, Luis A. Caffarelli, and Avner Friedman, Compressible flows of jets and cavities, J. Differential Equations 56 (1985), no. 1, 82–141. MR 772122, DOI 10.1016/0022-0396(85)90101-9
Thierry Aubin, Some nonlinear problems in Riemannian geometry, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. MR 1636569, DOI 10.1007/978-3-662-13006-3
Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
John W. Green, Harmonic functions in domains with multiple boundary points, Amer. J. Math. 61 (1939), 609–632. MR 90, DOI 10.2307/2371316
Luis A. Caffarelli and Xavier Cabré, Fully nonlinear elliptic equations, American Mathematical Society Colloquium Publications, vol. 43, American Mathematical Society, Providence, RI, 1995. MR 1351007, DOI 10.1090/coll/043
Sunčica Čanić, Barbara Lee Keyfitz, and Gary M. Lieberman, A proof of existence of perturbed steady transonic shocks via a free boundary problem, Comm. Pure Appl. Math. 53 (2000), no. 4, 484–511. MR 1733695, DOI 10.1002/(SICI)1097-0312(200004)53:4<484::AID-CPA3>3.3.CO;2-B
Gui-Qiang Chen and James Glimm, Global solutions to the compressible Euler equations with geometrical structure, Comm. Math. Phys. 180 (1996), no. 1, 153–193. MR 1403862, DOI 10.1007/BF02101185
Shuxing Chen, Existence of stationary supersonic flows past a pointed body, Arch. Ration. Mech. Anal. 156 (2001), no. 2, 141–181. MR 1814974, DOI 10.1007/s002050100121
Shuxing Chen, Asymptotic behaviour of supersonic flow past a convex combined wedge, Chinese Ann. Math. Ser. B 19 (1998), no. 3, 255–264. A Chinese summary appears in Chinese Ann. Math. Ser. A 19 (1998), no. 4, 533. MR 1667344
Morgan Ward, Ring homomorphisms which are also lattice homomorphisms, Amer. J. Math. 61 (1939), 783–787. MR 10, DOI 10.2307/2371336
T. N. E. Greville, Some extensions of Mr. Beers’s method of interpolation, Record. Amer. Inst. Actuar. 34 (1945), 188–193. MR 20001
Guang Chang Dong, Nonlinear partial differential equations of second order, Translations of Mathematical Monographs, vol. 95, American Mathematical Society, Providence, RI, 1991. Translated from the Chinese by Kai Seng Chou [Kaising Tso]. MR 1134129, DOI 10.1090/mmono/095
Saunders MacLane and O. F. G. Schilling, Infinite number fields with Noether ideal theories, Amer. J. Math. 61 (1939), 771–782. MR 19, DOI 10.2307/2371335
David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190, DOI 10.1007/978-3-642-61798-0
James Glimm and Andrew J. Majda (eds.), Multidimensional hyperbolic problems and computations, The IMA Volumes in Mathematics and its Applications, vol. 29, Springer-Verlag, New York, 1991. Papers from the IMA Workshop held in Minneapolis, Minnesota, April 3–14, 1989. MR 1087068, DOI 10.1007/978-1-4613-9121-0

Fudan J.

7

Carlos E. Kenig, Harmonic analysis techniques for second order elliptic boundary value problems, CBMS Regional Conference Series in Mathematics, vol. 83, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1994. MR 1282720, DOI 10.1090/cbms/083
Peter D. Lax, Hyperbolic systems of conservation laws and the mathematical theory of shock waves, Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 11, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1973. MR 0350216
Da Qian Li, On a free boundary problem, Chinese Ann. Math. 1 (1980), no. 3-4, 351–358 (English, with Chinese summary). MR 619582
Gary M. Lieberman, Regularity of solutions of nonlinear elliptic boundary value problems, J. Reine Angew. Math. 369 (1986), 1–13. MR 850625, DOI 10.1515/crll.1986.369.1
Gary M. Lieberman and Neil S. Trudinger, Nonlinear oblique boundary value problems for nonlinear elliptic equations, Trans. Amer. Math. Soc. 295 (1986), no. 2, 509–546. MR 833695, DOI 10.1090/S0002-9947-1986-0833695-6
V. I. Ivanov, A mathematical model of the Stokes soliton, Izv. Ross. Akad. Nauk Mekh. Zhidk. Gaza 1 (1999), 174–180 (Russian, with Russian summary); English transl., Fluid Dynam. 34 (1999), no. 1, 147–152. MR 1697990, DOI 10.1007/BF02698766
Rudolph E. Langer, The boundary problem of an ordinary linear differential system in the complex domain, Trans. Amer. Math. Soc. 46 (1939), 151–190 and Correction, 467 (1939). MR 84, DOI 10.1090/S0002-9947-1939-0000084-7
Saunders MacLane, Steinitz field towers for modular fields, Trans. Amer. Math. Soc. 46 (1939), 23–45. MR 17, DOI 10.1090/S0002-9947-1939-0000017-3
Cathleen S. Morawetz, Potential theory for regular and Mach reflection of a shock at a wedge, Comm. Pure Appl. Math. 47 (1994), no. 5, 593–624. MR 1278346, DOI 10.1002/cpa.3160470502
David G. Schaeffer, Supersonic flow past a nearly straight wedge, Duke Math. J. 43 (1976), no. 3, 637–670. MR 413736
P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14
Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
Yongqian Zhang, Global existence of steady supersonic potential flow past a curved wedge with a piecewise smooth boundary, SIAM J. Math. Anal. 31 (1999), no. 1, 166–183. MR 1742300, DOI 10.1137/S0036141097331056