Free boundary problems and transonic shocks for the Euler equations in unbounded domains
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 3 (2004) no. 4, p. 827-869

We establish the existence and stability of multidimensional transonic shocks (hyperbolic-elliptic shocks), which are not nearly orthogonal to the flow direction, for the Euler equations for steady compressible potential fluids in unbounded domains in ${ℝ}^{n},n\ge 3$. The Euler equations can be written as a second order nonlinear equation of mixed hyperbolic-elliptic 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 multidimensional transonic shock which divides two regions of ${C}^{2,\alpha }$ flow, and the equation is hyperbolic in the upstream region where the ${C}^{2,\alpha }$ perturbed flow is supersonic. In this paper, we develop a new approach to deal with such free boundary problems and establish the existence and stability of multidimensional transonic shocks near planes. We first reformulate the free boundary problem into a fixed conormal boundary value problem for a nonlinear elliptic equation of second order in unbounded domains and then develop techniques to solve this elliptic problem. Our results indicate that there exists a solution of the free boundary problem such that the equation is always elliptic in the unbounded downstream region, the uniform velocity state at infinity in the downstream direction is equal to the unperturbed downstream velocity state, and the free boundary is ${C}^{2,\alpha }$, provided that the hyperbolic phase is close in ${C}^{2,\alpha }$ to a uniform flow. We further prove that the free boundary is stable under the ${C}^{2,\alpha }$ steady perturbation of the hyperbolic phase. Moreover, we extend our existence results to the case that the regularity of the steady perturbation is only ${C}^{1,1}$, and we introduce another simpler approach to deal with the existence and stability problem when the regularity of the steady perturbation is ${C}^{3,\alpha }$ or higher. We also establish the existence and stability of multidimensional transonic shocks near spheres in ${ℝ}^{n}$.

Classification:  35M10,  35J65,  35R35,  76L05,  76H05,  35B45,  35B35
@article{ASNSP_2004_5_3_4_827_0,
author = {Chen, Gui-Qiang and Feldman, Mikhail},
title = {Free boundary problems and transonic shocks for the Euler equations in unbounded domains},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 3},
number = {4},
year = {2004},
pages = {827-869},
zbl = {1170.35483},
mrnumber = {2124589},
language = {en},
url = {http://www.numdam.org/item/ASNSP_2004_5_3_4_827_0}
}

Chen, Gui-Qiang; Feldman, Mikhail. Free boundary problems and transonic shocks for the Euler equations in unbounded domains. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 3 (2004) no. 4, pp. 827-869. http://www.numdam.org/item/ASNSP_2004_5_3_4_827_0/

 H. W. Alt - L. A. Caffarelli, Existence and regularity for a minimum problem with free boundary, J. Reine Angew. Math. 325 (1981), 105-144. | MR 618549 | Zbl 0449.35105

 H. W. Alt - L. A. Caffarelli - A. Friedman, A free-boundary problem for quasilinear elliptic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 11 (1984), 1-44. | Numdam | MR 752578 | Zbl 0554.35129

 H. W. Alt - L. A. Caffarelli - A. Friedman, Compressible flows of jets and cavities, J. Differential Equations 56 (1985), 82-141. | MR 772122 | Zbl 0614.76074

 L. Bers, Existence and uniqueness of subsonic flows past a given profile, Comm. Pure Appl. Math. 7 (1954), 441-504. | MR 65334 | Zbl 0058.40601

 S. Canić - B. L. Keyfitz - G. Lieberman, A proof of existence of perturbed steady transonic shocks via a free boundary problem, Comm. Pure Appl. Math. 53 (2000), 484-511. | MR 1733695 | Zbl 1017.76040

 G.-Q. Chen - M. Feldman, Multidimensional transonic shocks and free boundary problems for nonlinear equations of mixed type, J. Amer. Math. Soc. 16 (2003), 461-494. | MR 1969202 | Zbl 1015.35075

 G.-Q. Chen - M. Feldman, Steady transonic shocks and free boundary problems in infinite cylinders for the Euler equations, Comm. Pure Appl. Math. 57 (2004), 310-356. | MR 2020107 | Zbl 1075.76036

 S.-X. Chen, Existence of stationary supersonic flows past a point body, Arch. Rational Mech. Anal. 156 (2001), 141-181. | MR 1814974 | Zbl 0979.76041

 S.-X. Chen, Asymptotic behavior of supersonic flow past a convex combined wedge, Chinese Ann. Math. 19B (1998), 255-264. | MR 1667344 | Zbl 0914.35098

 R. Courant - K. O. Friedrichs, “Supersonic Flow and Shock Waves”, Springer-Verlag, New York, 1948. | MR 421279 | Zbl 0365.76001

 C. M. Dafermos, “Hyperbolic Conservation Laws in Continuum Physics”, Springer-Verlag, Berlin, 2000. | MR 1763936 | Zbl 0940.35002

 G. Dong, “Nonlinear Partial Differential Equations of Second Order”, Transl. Math. Monographs, 95, AMS, Providence, RI, 1991. | MR 1134129 | Zbl 0759.35001

 L. C. Evans - W. Gangbo, Differential equations methods for the Monge-Kantorovich mass transfer problem, Mem. Amer. Math. Soc. n. 653, 137 (1999). | MR 1464149 | Zbl 0920.49004

 R. Finn - D. Gilbarg, Asymptotic behavior and uniqueness of plane subsonic flows, Comm. Pure Appl. Math. 10 (1957), 23-63. | MR 86556 | Zbl 0077.18801

 R. Finn - D. Gilbarg, Three-dimensional subsonic flows, and asymptotic estimates for elliptic partial differential equations, Acta Math. 98 (1957), 265-296. | MR 92912 | Zbl 0078.40001

 D. Gilbarg - N. Trudinger, “Elliptic Partial Differential Equations of Second Order”, 2nd Ed., Springer-Verlag, Berlin, 1983. | MR 737190 | Zbl 0562.35001

 J. Glimm - A. Majda, “Multidimensional Hyperbolic Problems and Computations”, Springer-Verlag, New York, 1991. | MR 1087068 | Zbl 0718.00008

 C.-H. Gu, A method for solving the supersonic flow past a curved wedge (in Chinese), J. Fudan Univ. Nat. Sci. 7 (1962), 11-14.

 D. Kinderlehrer - L. Nirenberg, Regularity in free boundary problems, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 4 (1977), 373-391. | Numdam | MR 440187 | Zbl 0352.35023

 P. D. Lax, “Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves”, CBMS-RCSM, SIAM, Philiadelphia, 1973. | MR 350216 | Zbl 0268.35062

 T.-T. Li, On a free boundary problem, Chinese Ann. Math. 1 (1980), 351-358. | MR 619582

 G. Lieberman, Regularity of solutions of nonlinear elliptic boundary value problems, J. Reine Angew. Math. 369 (1986), 1-13. | MR 850625 | Zbl 0585.35014

 G. Lieberman, Hölder continuity of the gradient of solutions of uniformly parabolic equations with conormal boundary conditions, Ann. Mat. Pura Appl. (4) 148 (1987), 77-99. | MR 932759 | Zbl 0658.35050

 G. Lieberman, Mixed boundary value problems for elliptic and parabolic differential equations of second order, J. Math. Anal. Appl. n. 2, 113 (1986), 422-440. | MR 826642 | Zbl 0609.35021

 G. Lieberman - N. Trudinger, Nonlinear oblique boundary value problems for nonlinear elliptic equations, Trans. Amer. Math. Soc. 295 (1986), 509-546. | MR 833695 | Zbl 0619.35047

 W.-C. Lien - T.-P. Liu, Nonlinear stability of a self-similar 3-dimensional gas flow, Comm. Math. Phys. 204 (1999), 525-549. | MR 1707631 | Zbl 0945.76033

 A. Majda, “The stability of multidimensional shock fronts”, Mem. Amer. Math. Soc. n. 275, 41, AMS, Providence, 1983. | MR 683422 | Zbl 0506.76075

 A. Majda, “The existence of multidimensional shock fronts”, Mem. Amer. Math. Soc. n. 281, 43, AMS, Providence, 1983. | MR 699241 | Zbl 0517.76068

 G. Métivier, Stability of multi-dimensional weak shocks, Comm. Partial Differential Equations 15 (1990), 983-1028. | MR 1070236 | Zbl 0711.35078

 C. S. Morawetz, On the non-existence of continuous transonic flows past profiles I-III, Comm. Pure Appl. Math. 9 (1956), 45-68; 10 (1957), 107-131; 11 (1958), 129-144. | MR 78130 | Zbl 0077.18901

 O. Oleinik, “Some Asymptotic Problems in the Theory of Partial Differential Equations”, Lezioni Lincee. [Lincei Lectures] Cambridge University Press, Cambridge, 1996. | MR 1410755 | Zbl 1075.35500

 D. G. Schaeffer, Supersonic flow past a nearly straight wedge, Duke Math. J. 43 (1976), 637-670. | MR 413736 | Zbl 0356.76046

 M. Shiffman, On the existence of subsonic flows of a compressible fluid, J. Rational Mech. Anal. 1 (1952), 605-652. | MR 51651 | Zbl 0048.19301

 Y. Zhang, Steady supersonic flow past an almost straight wedge with large vertex angle, J. Differential Equations 192 (2003), 1-46. | MR 1987082 | Zbl 1035.35079