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Transonic flows with shocks past curved wedges for the full Euler equations
1. | Mathematical Institute, University of Oxford, Andrew Wiles Building, Radclie Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, United Kingdom |
2. | Department of Mathematics, Southern University of Science and Technology, Shenzhen, Guangdong 518055, China |
3. | Department of Mathematics, University of Wisconsin-Madison, Madison, WI 53706-1388, United States |
References:
[1] |
G.-Q. Chen, J. Chen and M. Feldman, Transonic shocks and free boundary problems for the full Euler equations in infinite nozzles, J. Math. Pures Appl. (9), 88 (2007), 191-218.
doi: 10.1016/j.matpur.2007.04.008. |
[2] |
G.-Q. Chen and B. Fang, Stability of transonic shock-fronts in three-dimensional conical steady potential flow past a perturbed cone, Discrete Contin. Dyn. Syst., 23 (2009), 85-114.
doi: 10.3934/dcds.2009.23.85. |
[3] |
G.-Q. Chen and M. Feldman, Multidimensional transonic shocks and free boundary problems for nonlinear equations of mixed type, J. Amer. Math. Soc., 16 (2003), 461-494.
doi: 10.1090/S0894-0347-03-00422-3. |
[4] |
G.-Q. Chen and T. Li, Well-posedness for two-dimnsional steady supersonic Euler flows past a Lipschitz wedge, J. Diff. Eqs., 244 (2008), 1521-1550. |
[5] |
G.-Q. Chen, Y.-Q. Zhang and D.-W. Zhu, Existence and stability of supersonic Euler flows past Lipschitz wedges, Arch. Rational Mech. Anal., 181 (2006), 261-310.
doi: 10.1007/s00205-005-0412-3. |
[6] |
S.-X. Chen, Supersonic flow past a concave wedge, Science in China, 10 (1997), 903-910. |
[7] |
S.-X. Chen, Asymptotic behavior of supersonic flow past a convex combined wedge, Chin. Ann. Math., 19 (1998), 255-264. |
[8] |
S.-X. Chen, A free boundary problem of elliptic equation arising in supersonic flow past a conical body, Z. Angew. Math. Phys., 54 (2003), 387-409.
doi: 10.1007/s00033-003-2111-y. |
[9] |
S.-X. Chen, Stability of transonic shock front in multi-dimensional Euler system, Trans. Amer. Math. Soc., 357 (2005), 287-308.
doi: 10.1090/S0002-9947-04-03698-0. |
[10] |
S.-X. Chen, Stability of a Mach configuration, Comm. Pure Appl. Math., 59 (2006), 1-35.
doi: 10.1002/cpa.20108. |
[11] |
S.-X. Chen and B.-X. Fang, Stability of transonic shocks in supersonic flow past a wedge, J. Diff. Eqs., 233 (2007), 105-135.
doi: 10.1016/j.jde.2006.09.020. |
[12] |
S.-X. Chen, Z. Xin and H. Yin, Global shock waves for the supersonic flow past a perturbed cone, Comm. Math. Phys., 228 (2002), 47-84.
doi: 10.1007/s002200200652. |
[13] |
S.-X. Chen and H. Yuan, Transonic shocks in compressible flow assing a duct for three-dimensional Euler system, Arch. Rational Mech. Anal., 187 (2008), 523-556.
doi: 10.1007/s00205-007-0079-z. |
[14] |
A. Chorin and J. A. Marsden, A Mathematical Introduction to Fluid Mechanics, $3^{rd}$ edition, Springer-Verlag, New York, 1993. |
[15] |
R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, Wiley Interscience, New York, 1948. |
[16] |
B.-X. Fang, Stability of transonic shocks for the full Euler system in supersonic flow past a wedge, Math. Methods Appl. Sci., 29 (2006), 1-26.
doi: 10.1002/mma.661. |
[17] |
D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, $2^{nd}$ edition, Springer-Verlag, Berlin, 1983.
doi: 10.1007/978-3-642-61798-0. |
[18] |
J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math., 18 (1965), 697-715.
doi: 10.1002/cpa.3160180408. |
[19] |
C. Gu, A method for solving the supersonic flow past a curved wedge, Fudan J.(Nature Sci.), 7 (1962), 11-14. |
[20] |
P. D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, CBMS-RCSAM, 11. SIAM, Philadelphia, Pa., 1973. |
[21] |
P. D. Lax, Hyperbolic systems of conservation laws in several space variables, In: Current Topics in Partial Differential Equations, 327-341, Kinokuniya, Tokyo, 1986. |
[22] |
P. D. Lax and X.-D. Liu, Solution of two-dimensional Riemann problems of gas dynamics by positive schemes, SIAM J. Sci. Comput., 19 (1998), 319-340.
doi: 10.1137/S1064827595291819. |
[23] |
T. Li, On a free boundary problem, Chinese Ann. Math., 1 (1980), 351-358. |
[24] |
W. Lien and T.-P. Li, Nonlinear stability of a self-similar 3-dimensional gas flow, Commun. Math. Phys., 204 (1999), 525-549.
doi: 10.1007/s002200050656. |
[25] |
A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Springer-Verlag, New York, 1984.
doi: 10.1007/978-1-4612-1116-7. |
[26] |
L. Prandtl, Allgemeine Überlegungen über die Strömung zusammendrückbarer Flüssigkeiten, Z. Angew. Math. Mech., 16 (1936), 129-142. |
[27] |
D. G. Schaeffer, Supersonic flow past a nearly straight wedge, Duke Math. J., 43 (1976), 637-670.
doi: 10.1215/S0012-7094-76-04351-9. |
[28] |
J. von Neumann, Discussion on the existence and uniqueness or multiplicity of solutions of the aerodynamical equations [Reprinted from MR0044302], Bull. Amer. Math. Soc. (N.S.), 47 (2010), 145-154.
doi: 10.1090/S0273-0979-09-01281-6. |
[29] |
Y. Zhang, Steady supersonic flow past an almost straight wedge with large vertex angle, J. Diff. Eqs., 192 (2003), 1-46.
doi: 10.1016/S0022-0396(03)00037-8. |
show all references
References:
[1] |
G.-Q. Chen, J. Chen and M. Feldman, Transonic shocks and free boundary problems for the full Euler equations in infinite nozzles, J. Math. Pures Appl. (9), 88 (2007), 191-218.
doi: 10.1016/j.matpur.2007.04.008. |
[2] |
G.-Q. Chen and B. Fang, Stability of transonic shock-fronts in three-dimensional conical steady potential flow past a perturbed cone, Discrete Contin. Dyn. Syst., 23 (2009), 85-114.
doi: 10.3934/dcds.2009.23.85. |
[3] |
G.-Q. Chen and M. Feldman, Multidimensional transonic shocks and free boundary problems for nonlinear equations of mixed type, J. Amer. Math. Soc., 16 (2003), 461-494.
doi: 10.1090/S0894-0347-03-00422-3. |
[4] |
G.-Q. Chen and T. Li, Well-posedness for two-dimnsional steady supersonic Euler flows past a Lipschitz wedge, J. Diff. Eqs., 244 (2008), 1521-1550. |
[5] |
G.-Q. Chen, Y.-Q. Zhang and D.-W. Zhu, Existence and stability of supersonic Euler flows past Lipschitz wedges, Arch. Rational Mech. Anal., 181 (2006), 261-310.
doi: 10.1007/s00205-005-0412-3. |
[6] |
S.-X. Chen, Supersonic flow past a concave wedge, Science in China, 10 (1997), 903-910. |
[7] |
S.-X. Chen, Asymptotic behavior of supersonic flow past a convex combined wedge, Chin. Ann. Math., 19 (1998), 255-264. |
[8] |
S.-X. Chen, A free boundary problem of elliptic equation arising in supersonic flow past a conical body, Z. Angew. Math. Phys., 54 (2003), 387-409.
doi: 10.1007/s00033-003-2111-y. |
[9] |
S.-X. Chen, Stability of transonic shock front in multi-dimensional Euler system, Trans. Amer. Math. Soc., 357 (2005), 287-308.
doi: 10.1090/S0002-9947-04-03698-0. |
[10] |
S.-X. Chen, Stability of a Mach configuration, Comm. Pure Appl. Math., 59 (2006), 1-35.
doi: 10.1002/cpa.20108. |
[11] |
S.-X. Chen and B.-X. Fang, Stability of transonic shocks in supersonic flow past a wedge, J. Diff. Eqs., 233 (2007), 105-135.
doi: 10.1016/j.jde.2006.09.020. |
[12] |
S.-X. Chen, Z. Xin and H. Yin, Global shock waves for the supersonic flow past a perturbed cone, Comm. Math. Phys., 228 (2002), 47-84.
doi: 10.1007/s002200200652. |
[13] |
S.-X. Chen and H. Yuan, Transonic shocks in compressible flow assing a duct for three-dimensional Euler system, Arch. Rational Mech. Anal., 187 (2008), 523-556.
doi: 10.1007/s00205-007-0079-z. |
[14] |
A. Chorin and J. A. Marsden, A Mathematical Introduction to Fluid Mechanics, $3^{rd}$ edition, Springer-Verlag, New York, 1993. |
[15] |
R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, Wiley Interscience, New York, 1948. |
[16] |
B.-X. Fang, Stability of transonic shocks for the full Euler system in supersonic flow past a wedge, Math. Methods Appl. Sci., 29 (2006), 1-26.
doi: 10.1002/mma.661. |
[17] |
D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, $2^{nd}$ edition, Springer-Verlag, Berlin, 1983.
doi: 10.1007/978-3-642-61798-0. |
[18] |
J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math., 18 (1965), 697-715.
doi: 10.1002/cpa.3160180408. |
[19] |
C. Gu, A method for solving the supersonic flow past a curved wedge, Fudan J.(Nature Sci.), 7 (1962), 11-14. |
[20] |
P. D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, CBMS-RCSAM, 11. SIAM, Philadelphia, Pa., 1973. |
[21] |
P. D. Lax, Hyperbolic systems of conservation laws in several space variables, In: Current Topics in Partial Differential Equations, 327-341, Kinokuniya, Tokyo, 1986. |
[22] |
P. D. Lax and X.-D. Liu, Solution of two-dimensional Riemann problems of gas dynamics by positive schemes, SIAM J. Sci. Comput., 19 (1998), 319-340.
doi: 10.1137/S1064827595291819. |
[23] |
T. Li, On a free boundary problem, Chinese Ann. Math., 1 (1980), 351-358. |
[24] |
W. Lien and T.-P. Li, Nonlinear stability of a self-similar 3-dimensional gas flow, Commun. Math. Phys., 204 (1999), 525-549.
doi: 10.1007/s002200050656. |
[25] |
A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Springer-Verlag, New York, 1984.
doi: 10.1007/978-1-4612-1116-7. |
[26] |
L. Prandtl, Allgemeine Überlegungen über die Strömung zusammendrückbarer Flüssigkeiten, Z. Angew. Math. Mech., 16 (1936), 129-142. |
[27] |
D. G. Schaeffer, Supersonic flow past a nearly straight wedge, Duke Math. J., 43 (1976), 637-670.
doi: 10.1215/S0012-7094-76-04351-9. |
[28] |
J. von Neumann, Discussion on the existence and uniqueness or multiplicity of solutions of the aerodynamical equations [Reprinted from MR0044302], Bull. Amer. Math. Soc. (N.S.), 47 (2010), 145-154.
doi: 10.1090/S0273-0979-09-01281-6. |
[29] |
Y. Zhang, Steady supersonic flow past an almost straight wedge with large vertex angle, J. Diff. Eqs., 192 (2003), 1-46.
doi: 10.1016/S0022-0396(03)00037-8. |
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