×

Two dimensional subsonic Euler flows past a wall or a symmetric body. (English) Zbl 1338.35342

Summary: The existence and uniqueness of two dimensional steady compressible Euler flows past a wall or a symmetric body are established. More precisely, given positive convex horizontal velocity in the upstream, there exists a critical value \(\rho_{\mathrm{cr}}\) such that if the incoming density in the upstream is larger than \(\rho_{\mathrm{cr}}\), then there exists a subsonic flow past a wall. Furthermore, \(\rho_{\mathrm{cr}}\) is critical in the sense that there is no such subsonic flow if the density of the incoming flow is less than \(\rho_{\mathrm{cr}}\). The subsonic flows possess large vorticity and positive horizontal velocity above the wall except at the corner points on the boundary. Moreover, the existence and uniqueness of a two dimensional subsonic Euler flow past a symmetric body are also obtained when the incoming velocity field is a general small perturbation of a constant velocity field and the density of the incoming flow is larger than a critical value. The asymptotic behavior of the flows is obtained with the aid of some integral estimates for the difference between the velocity field and its far field states.

MSC:

35Q31 Euler equations
35B40 Asymptotic behavior of solutions to PDEs
76G25 General aerodynamics and subsonic flows
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Bers, L.: Mathematical Aspects of Subsonic and Transonic Gas Dynamics, Surveys in Applied Mathematics, Vol. 3. Wiley, New York, 1958 · Zbl 0083.20501
[2] Bers L.: Existence and uniqueness of a subsonic flow past a given profile. Commun. Pure Appl. Math. 7, 441-504 (1954) · Zbl 0058.40601 · doi:10.1002/cpa.3160070303
[3] Chen C., Xie C.J.: Existence of steady subsonic Euler flows through infinitely long periodic nozzles. J. Differ. Equ. 252, 4315-4331 (2012) · Zbl 1402.35168 · doi:10.1016/j.jde.2011.12.015
[4] Chen C., Xie C.J.: Three dimensional steady subsonic Euler flows in bounded nozzles. J. Differ. Equ. 256, 3684-3708 (2014) · Zbl 1457.35030 · doi:10.1016/j.jde.2014.02.016
[5] Chen G.Q., Dafermos C., Slemrod M., Wang D.H.: On two-dimensional sonic-subsonic flow. Commun. Math. Phys. 271, 635-647 (2007) · Zbl 1128.35070 · doi:10.1007/s00220-007-0211-9
[6] Chen G.Q., Deng X.M., Xiang W.: Global steady subsonic flows through infinitely long nozzles for the full Euler equations. SIAM J. Math. Anal. 44, 2888-2919 (2012) · Zbl 1298.35138 · doi:10.1137/11085325X
[7] Chen G.Q., Feldman M.: Multidimensional transonic shocks and free boundary problems for nonlinear equations of mixted type. J. Am. Math. Soc. 16, 461-494 (2003) · Zbl 1015.35075 · doi:10.1090/S0894-0347-03-00422-3
[8] Chen G.Q., Huang F.M., Wang T.Y.: Sonic-subsonic limit of approximate solutions to multidimensional steady Euler equations. Arch. Rational Mech. Anal. 219, 719-740 (2016) · Zbl 1333.35168 · doi:10.1007/s00205-015-0905-7
[9] Chen J.: Subsonic flows for the full Euler equations in half plane. J. Hyperbolic Differ. Equ. 6, 207-228 (2009) · Zbl 1180.35419 · doi:10.1142/S0219891609001873
[10] Chen S.X., Yuan H.R.: Transonic shocks in compressible flow passing a duct for three-dimesional Euler systems. Arch. Rational Mech. Anal. 187, 523-556 (2008) · Zbl 1140.76015 · doi:10.1007/s00205-007-0079-z
[11] Dong G.C., Ou B.: Subsonic flows around a body in space. Commun. Partial Differ. Equ. 18, 355-379 (1993) · Zbl 0813.35013 · doi:10.1080/03605309308820933
[12] Du L.L., Duan B.: Global subsonic Euler flows in an infinitely long axisymmetric nozzle. J. Differ. Equ. 250, 813-847 (2011) · Zbl 1207.35244 · doi:10.1016/j.jde.2010.06.005
[13] Du L.L., Duan B.: Note on the uniqueness of subsonic Euler flows in an axisymmetric nozzle. Appl. Math. Lett. 25, 153-156 (2012) · Zbl 1277.76006 · doi:10.1016/j.aml.2011.07.020
[14] Du L.L., Weng S.K., Xin Z.P.: Subsonic irrotational flows in a finitely long nozzle with variable end pressure. Commun. Partial Differ. Equ. 39, 666-695 (2014) · Zbl 1306.76023 · doi:10.1080/03605302.2013.873938
[15] Du L.L., Xie C.J.: On subsonic Euler flows with stagnation points in two dimensional nozzles. Indiana Univ. Math. J. 63, 1499-1523 (2014) · Zbl 1315.35153 · doi:10.1512/iumj.2014.63.5366
[16] Du L.L., Xie C.J., Xin Z.P.: Steady subsonic ideal flows through an infinitely long nozzle with large vorticity. Commun. Math. Phys. 328, 327-354 (2014) · Zbl 1293.35224 · doi:10.1007/s00220-014-1951-y
[17] Du L.L., Xin Z.P., Yan W.: Subsonic flows in a multi-dimensional nozzle. Arch. Rational Mech. Anal. 201, 965-1012 (2011) · Zbl 1452.76200 · doi:10.1007/s00205-011-0406-2
[18] Duan B., Luo Z.: Three-dimensional full Euler flows in axisymmetric nozzles. J. Differ. Equ. 254, 2705-2731 (2013) · Zbl 1308.76230 · doi:10.1016/j.jde.2013.01.008
[19] Evans. L.C.: Partial differential equations, Graduate Studies in Mathematics, Vol. 19. American Mathematical Society, Providence, 1998 · Zbl 0902.35002
[20] Finn R., Gilbarg D.: Asymptotic behavior and uniqueness of plane subsonic flows. Commun. Pure Appl. Math. 10, 23-63 (1957) · Zbl 0078.40001 · doi:10.1002/cpa.3160100102
[21] Finn R., Gilbarg D.: Three-dimensional subsonic flows, and asymptotic estimates for elliptic partial differential equations. Acta Math. 98, 265-296 (1957) · Zbl 0078.40001 · doi:10.1007/BF02404476
[22] Gilbarg D., Shiffman M.: On bodies achieving extreme values of the critical Mach number. I. J. Rational Mech. Anal. 3, 209-230 (1954) · Zbl 0055.18801
[23] Gilbarg D., Trudinger N.S.: Elliptic Partial Differential Equations of Second Order, Classics in Mathematics. Springer, Berlin (2001) · Zbl 1042.35002
[24] Huang F.M., Wang T.Y., Wang Y.: On multi-dimensional sonic-subsonic flow. Acta Math. Sci. 31, 2131-2140 (2011) · Zbl 1265.76039 · doi:10.1016/S0252-9602(11)60389-5
[25] Li J., Xin Z.P., Yin H.C.: On transonic shocks in a nozzle with variable end pressures. Commun. Math. Phys. 291, 111-150 (2009) · Zbl 1187.35138 · doi:10.1007/s00220-009-0870-9
[26] Li J., Xin Z.P., Yin H.C.: Transonic shocks for the full compressible Euler system in a general two-dimensional de Laval nozzle. Arch. Rational Mech. Anal. 207, 533-581 (2013) · Zbl 1320.76056 · doi:10.1007/s00205-012-0580-x
[27] Morawetz C.S.: On the non-existence of continuous transonic flows past profiles. I. Commun. Pure Appl. Math. 9, 5-68 (1956) · Zbl 0070.20206 · doi:10.1002/cpa.3160090104
[28] Morawetz C.S.: On the non-existence of continuous transonic flows past profiles. II. Commun. Pure Appl. Math. 10, 107-131 (1957) · Zbl 0077.18901 · doi:10.1002/cpa.3160100105
[29] Morawetz C.S.: On the non-existence of continuous transonic flows past profiles. III. Commun. Pure Appl. Math. 11, 129-144 (1958) · doi:10.1002/cpa.3160110107
[30] Nadirasvili N.S.: Lemma on the interior derivative and uniqueness of the solution of the second boundary value problem for second-order elliptic equations. Soviet Math. Dokl 24, 598-601 (1981) · Zbl 0509.35023
[31] Shiffman M.: On the existence of subsonic flows of a compressible fluid. J. Rational Mech. Anal. 1, 605-652 (1952) · Zbl 0048.19301
[32] Wang C.P., Xin Z.P.: Optimal Holder continuity for a class of degenarate elliptic problems with an application to subsonic-sonic flows. Commun. Partial Differ. Equ. 36, 873-924 (2011) · Zbl 1245.76132 · doi:10.1080/03605302.2010.535074
[33] Wang C.P., Xin Z.P.: On a degenerate free boundary problem and continuous subsonic-sonic flows in a convergent nozzle. Arch. Rational Mech. Anal. 208, 911-975 (2012) · Zbl 1286.35209 · doi:10.1007/s00205-012-0607-3
[34] Wang, C.P., Xin, Z.P.: Smooth Transonic Flows in De Laval Nozzles. arXiv:1304.2473 · Zbl 1414.35176
[35] Weng S.K.: Subsonic irrotational flows in a two-dimensional finitely long curved nozzle. Zeitschrift für Angewandte Mathematik und Physik 65, 203-220 (2014) · Zbl 1293.35261 · doi:10.1007/s00033-013-0318-0
[36] Weng, S.K.: A new formulation for the 3-D Euler equations with an application to subsonic flows in a cylinder. arXiv:1212.1635 · Zbl 1334.35218
[37] Xie C.J., Xin Z.P.: Global subsonic and subsonic-sonic flows through infinitely long nozzles. Indiana Univ. Math. J. 56, 2991-3023 (2007) · Zbl 1156.35076 · doi:10.1512/iumj.2007.56.3108
[38] Xie C.J., Xin Z.P.: Global subsonic and subsonic-sonic flows through infinitely long axially symmetric nozzles. J. Differ. Equ. 248, 2657-2683 (2010) · Zbl 1193.35143 · doi:10.1016/j.jde.2010.02.007
[39] Xie C.J., Xin Z.P.: Existence of global steady subsonic Euler flows through infinitely long nozzle. SIAM J. Math. Anal. 42, 751-784 (2010) · Zbl 1218.35170 · doi:10.1137/09076667X
[40] Xin Z.P., Yin H.C.: Transonic shock in a nozzle, I. Two-dimensional case. Comm. Pure Appl. Math. 58, 999-1050 (2005) · Zbl 1076.76043 · doi:10.1002/cpa.20025
[41] Xin Z.P., Yin H.C.: The transonic shock in a nozzle, 2-D and 3-D complete Euler systems. J. Differ. Equ. 245, 1014-1085 (2008) · Zbl 1165.35031 · doi:10.1016/j.jde.2008.04.010
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.