×

3-d axisymmetric transonic shock solutions of the full Euler system in divergent nozzles. (English) Zbl 1464.35096

Summary: We establish the stability of 3-D axisymmetric transonic shock solutions of the steady full Euler system in divergent nozzles under small perturbations of an incoming radial supersonic flow and a constant pressure at the exit of the nozzles. To study 3-D axisymmetric transonic shock solutions of the full Euler system, we use a stream function formulation of the full Euler system for a 3-D axisymmetric flow. We resolve the singularity issue arising in stream function formulations of the full Euler system for a 3-D axisymmetric flow. We develop a new scheme to determine a shock location of a transonic shock solution of the steady full Euler system based on the stream function formulation.

MSC:

35J46 First-order elliptic systems
35Q31 Euler equations
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Ambrosio, L., Carlotto, A., Massaccesi, A.: Lectures on elliptic partial differential equations, Appunti. Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)]], vol. 18. Edizioni della Normale, Pisa 2018. doi:10.1007/978-88-7642-443-4 · Zbl 1416.35001
[2] Bae, M.; Duan, B.; Xie, C., Subsonic flow for the multidimensional Euler-Poisson system, Arch. Ration. Mech. Anal., 220, 1, 155-191 (2016) · Zbl 1339.35222 · doi:10.1007/s00205-015-0930-6
[3] Bae, M.; Feldman, M., Transonic shocks in multidimensional divergent nozzles, Arch. Ration. Mech. Anal., 201, 3, 777-840 (2011) · Zbl 1268.76053 · doi:10.1007/s00205-011-0424-0
[4] Bae, M.; Weng, S., 3-D axisymmetric subsonic flows with nonzero swirl for the compressible Euler-Poisson system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 35, 1, 161-186 (2018) · Zbl 1384.35022 · doi:10.1016/j.anihpc.2017.03.004
[5] Chen, GQ; Chen, J.; Feldman, M., Transonic shocks and free boundary problems for the full Euler equations in infinite nozzles, J. Math. Pures Appl. (9), 88, 2, 191-218 (2007) · Zbl 1131.35061 · doi:10.1016/j.matpur.2007.04.008
[6] Chen, GQ; Chen, J.; Song, K., Transonic nozzle flows and free boundary problems for the full Euler equations, J. Differ. Equ., 229, 1, 92-120 (2006) · Zbl 1142.35510 · doi:10.1016/j.jde.2006.04.015
[7] Chen, GQ; Feldman, M., Multidimensional transonic shocks and free boundary problems for nonlinear equations of mixed type, J. Am. Math. Soc., 16, 3, 461-494 (2003) · Zbl 1015.35075 · doi:10.1090/S0894-0347-03-00422-3
[8] Chen, GQ; Feldman, M., Steady transonic shocks and free boundary problems for the Euler equations in infinite cylinders, Commun. Pure Appl. Math., 57, 3, 310-356 (2004) · Zbl 1075.76036 · doi:10.1002/cpa.3042
[9] Chen, GQ; Feldman, M., Existence and stability of multidimensional transonic flows through an infinite nozzle of arbitrary cross-sections, Arch. Ration. Mech. Anal., 184, 2, 185-242 (2007) · Zbl 1190.76158 · doi:10.1007/s00205-006-0025-5
[10] Chen, GQG; Yuan, H., Local uniqueness of steady spherical transonic shock-fronts for the three-dimensional full Euler equations, Commun. Pure Appl. Anal., 12, 6, 2515-2542 (2013) · Zbl 1267.35004 · doi:10.3934/cpaa.2013.12.2515
[11] Chen, S., Stability of transonic shock fronts in two-dimensional Euler systems, Trans. Am. Math. Soc., 357, 1, 287-308 (2005) · Zbl 1077.35094 · doi:10.1090/S0002-9947-04-03698-0
[12] Chen, S., Transonic shocks in 3-D compressible flow passing a duct with a general section for Euler systems, Trans. Am. Math. Soc., 360, 10, 5265-5289 (2008) · Zbl 1158.35064 · doi:10.1090/S0002-9947-08-04493-0
[13] Chen, S.; Yuan, H., Transonic shocks in compressible flow passing a duct for three-dimensional Euler systems, Arch. Ration. Mech. Anal., 187, 3, 523-556 (2008) · Zbl 1140.76015 · doi:10.1007/s00205-007-0079-z
[14] Courant, R., Friedrichs, K.O.: Supersonic flow and shock waves. Springer-Verlag, New York-Heidelberg 1976. https://mathscinet.ams.org/mathscinet-getitem?mr=0421279. Reprinting of the 1948 original, Applied Mathematical Sciences, Vol. 21 · Zbl 0365.76001
[15] Giaquinta, M.: Multiple integrals in the calculus of variations and nonlinear elliptic systems, Annals of Mathematics Studies, vol. 105. Princeton University Press, Princeton, NJ 1983. https://mathscinet.ams.org/mathscinet-getitem?mr=717034 · Zbl 0516.49003
[16] Giaquinta, M., Martinazzi, L.: An introduction to the regularity theory for elliptic systems, harmonic maps and minimal graphs, Appunti. Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)], vol. 11, second edn. Edizioni della Normale, Pisa 2012. doi:10.1007/978-88-7642-443-4. https://mathscinet.ams.org/mathscinet-getitem?mr=3099262 · Zbl 1262.35001
[17] Han, Q., Lin, F.: Elliptic partial differential equations, Courant Lecture Notes in Mathematics, vol. 1, second edn. Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI 2011. https://mathscinet.ams.org/mathscinet-getitem?mr=2777537 · Zbl 1210.35031
[18] Li, J.; Xin, Z.; Yin, H., A free boundary value problem for the full Euler system and 2-D transonic shock in a large variable nozzle, Math. Res. Lett., 16, 5, 777-796 (2009) · Zbl 1194.35517 · doi:10.4310/MRL.2009.v16.n5.a3
[19] Li, J.; Xin, Z.; Yin, H., On transonic shocks in a nozzle with variable end pressures, Commun. Math. Phys., 291, 1, 111-150 (2009) · Zbl 1187.35138 · doi:10.1007/s00220-009-0870-9
[20] Li, J.; Xin, Z.; Yin, H., On transonic shocks in a conic divergent nozzle with axi-symmetric exit pressures, J. Differ. Equ., 248, 3, 423-469 (2010) · Zbl 1191.35179 · doi:10.1016/j.jde.2009.09.017
[21] Li, J.; Xin, Z.; Yin, H., Transonic shocks for the full compressible Euler system in a general two-dimensional de Laval nozzle, Arch. Ration. Mech. Anal., 207, 2, 533-581 (2013) · Zbl 1320.76056 · doi:10.1007/s00205-012-0580-x
[22] Liu, JG; Wang, WC, Characterization and regularity for axisymmetric solenoidal vector fields with application to Navier-Stokes equation, SIAM J. Math. Anal., 41, 5, 1825-1850 (2009) · Zbl 1197.35195 · doi:10.1137/080739744
[23] Liu, L.; Xu, G.; Yuan, H., Stability of spherically symmetric subsonic flows and transonic shocks under multidimensional perturbations, Adv. Math., 291, 696-757 (2016) · Zbl 1341.35105 · doi:10.1016/j.aim.2016.01.002
[24] Liu, L.; Yuan, H., Stability of cylindrical transonic shocks for the two-dimensional steady compressible Euler system, J. Hyperbolic Differ. Equ., 5, 2, 347-379 (2008) · Zbl 1158.35014 · doi:10.1142/S0219891608001519
[25] Xin, Z.; Yan, W.; Yin, H., Transonic shock problem for the Euler system in a nozzle, Arch. Ration. Mech. Anal., 194, 1, 1-47 (2009) · Zbl 1423.76386 · doi:10.1007/s00205-009-0251-8
[26] Xin, Z.; Yin, H., Transonic shock in a nozzle. I. Two-dimensional case, Commun. Pure Appl. Math., 58, 8, 999-1050 (2005) · Zbl 1076.76043 · doi:10.1002/cpa.20025
[27] Xin, Z.; Yin, H., Three-dimensional transonic shocks in a nozzle, Pac. J. Math., 236, 1, 139-193 (2008) · Zbl 1232.35097 · doi:10.2140/pjm.2008.236.139
[28] Xin, Z.; Yin, H., The transonic shock in a nozzle, 2-D and 3-D complete Euler systems, J. Differ. Equ., 245, 4, 1014-1085 (2008) · Zbl 1165.35031 · doi:10.1016/j.jde.2008.04.010
[29] Yuan, H., On transonic shocks in two-dimensional variable-area ducts for steady Euler system, SIAM J. Math. Anal., 38, 4, 1343-1370 (2006) · Zbl 1121.35081 · doi:10.1137/050642447
[30] Yuan, H., Transonic shocks for steady Euler flows with cylindrical symmetry, Nonlinear Anal., 66, 8, 1853-1878 (2007) · Zbl 1162.35429 · doi:10.1016/j.na.2006.02.045
[31] Yuan, H., A remark on determination of transonic shocks in divergent nozzles for steady compressible Euler flows, Nonlinear Anal. Real World Appl., 9, 2, 316-325 (2008) · Zbl 1137.76029 · doi:10.1016/j.nonrwa.2006.10.006
[32] Yuan, H., Zhao, Q.: Stabilization effect of frictions for transonic shocks in steady compressible euler flows passing three-dimensional ducts 2018
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.