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A Brezis-Nirenberg result for non-local critical equations in low dimension
Systems of singular integral equations and applications to existence of reversed flow solutions of Falkner-Skan equations
1. | Department of Computation Science, Chengdu University of Information Technology, Chengdu, Sichuan 610225, China |
2. | Department of Mathematics, Ryerson University, Toronto, Ontario, M5B 2K3, Canada |
References:
[1] |
R. P. Agarwal and D. O'Regan, Singular integral equations arising in Homann flow, Dyn. Cont. Discrete Impuls. Syst. Ser. B Appl. Algorithms, 9 (2002), 481-488. |
[2] |
D. J. Acheson, "Elementary Fluid Dynamics," Oxford Univ. Press, New York, 1990. |
[3] |
B. Brighi and J. D. Hoernel, On the concave and convex solutions of a mixed convection boundary layer approximation in a porous medium, Appl. Math. Lett., 19 (2006), 69-74.
doi: dx.doi.org/10.1016/j.aml.2005.02.038. |
[4] |
S. N. Brown and K. Stewartson, On the reversed flow solutions of the Falkner-Skan equations, Mathematika, 13 (1966), 1-6.
doi: dx.doi.org/10.1112/S0025579300004125. |
[5] |
W. A. Coppel, On a differential equation of boundary-layer theory, Phil. Trans. Roy. Soc. London, Ser. A, 253 (1960), 101-136.
doi: dx.doi.org/10.1112/S0025579300005052. |
[6] |
A. H. Craven and L. A. Peletier, On the uniqueness of solutions of the Falkner-Skan equation, Mathematika, 19 (1972), 129-133.
doi: dx.doi.org/10.1112/S0025579300005052. |
[7] |
I. G. Currie, "Fundamental Mechanics of Fluids," 3rd, Marcel Dekker, New York, 2003. |
[8] |
K. Deimling, "Nonlinear Functional Analysis," Springer-Verlag, Berlin, 1985.
doi: dx.doi.org/10.1007/978-3-662-00547-7. |
[9] |
M. Guedda and Z. Hammouch, On similarity and pseudo-similarity solutions of Falkner-Skan boundary layers, Fluid Dynam. Res., 38 (2006), 211-223.
doi: dx.doi.org/10.1016/j.fluiddyn.2005.11.001. |
[10] |
M. Guedda, Multiple solutions of mixed convenction boundary-layer approximations in a porous medium, Appl. Math. Lett., 19 (2006), 63-68.
doi: dx.doi.org/10.1016/j.aml.2005.02.037. |
[11] |
P. Hartman, "Ordinary Differential Equations," Classics in Applied Mathematics, 38, the Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. |
[12] |
P. Hartman, On the existence of similar solutions of some boundary layer problems, SIAM J. Math. Anal., 3 (1972), 120-147.
doi: dx.doi.org/10.1137/0503014. |
[13] |
S. P. Hastings, An existence theorem for a class of nonlinear boundary value problems including that of Falkner and Skan, J. Differential Equations, 9 (1971), 580-590.
doi: dx.doi.org/10.1016/0022-0396(71)90025-8. |
[14] |
S. P. Hastings, Reversed flow solutions of the Falkner-Skan equations, SIAM J. Appl. Math., 22 (1972), 329-334.
doi: dx.doi.org/10.1137/0122031. |
[15] |
K. Q. Lan and G. C. Yang, Positive solutions of the Falker-Skan equation arising in boundary layer theory, Canad. Math. Bull., 51 (2008), 386-398.
doi: dx.doi.org/10.4153/CMB-2008-039-7. |
[16] |
P. A. Libby and T. M. Liu, Further solutions of the Falkner-Skan equation, AIAA J., 5 (1967), 1040-1042.
doi: dx.doi.org/10.2514/3.4130. |
[17] |
T. Y. Na, "Computational Methods in Engineering Boundary Value Problems," Academic Press, 1979. |
[18] |
H. Schlichting and K. Gersten, "Boundary-Layer Theory," 8ed, Springer-Verlag, Berlin, 2000. |
[19] |
K. Schrader, A generalization of the Helly selection theorem, Bull. Amer. Math. Soc., 78 (1972), 415-419.
doi: dx.doi.org/10.1090/S0002-9904-1972-12923-9. |
[20] |
C. Sparrow and H. P. F. Swinnerton-Dyer, The Falkner-Skan equation. II. Dynamics and the bifurcations of $P$- and $Q$-orbits, J. Differential Equations, 183 (2002), 1-55.
doi: dx.doi.org/10.1006/jdeq.2001.4100. |
[21] |
K. Stewartson, Further solutions of the Falkner-Skan equation, Proc. Camb. Phil. Soc., 50 (1954), 454-465.
doi: dx.doi.org/10.1017/S030500410002956X. |
[22] |
H. P. F. Swinnerton-Dyer and C. T. Sparrow, The Falkner-Skan equation. I. The creation of strange invariant sets, J. Differential Equations, 119 (1995), 336-394.
doi: dx.doi.org/10.1006/jdeq.1995.1094. |
[23] |
K. K. Tam, A note on the existence of a solution of the Falkner-Skan equation, Canad. Math. Bull., 13 (1970), 125-127.
doi: dx.doi.org/10.4153/CMB-1970-026-8. |
[24] |
J. Wang, W. Gao and Z. Zhang, Singular nonlinear boundary value problems arising in boundary layer theory, J. Math. Anal. Appl., 233 (1999), 246-256.
doi: dx.doi.org/10.1006/jmaa.1999.6290. |
[25] |
H. Weyl, On the differential equations of the simplest boundary-layer problems, Ann. Math., 43 (1942), 381-407.
doi: dx.doi.org/10.2307/1968875. |
[26] |
G. C. Yang and K. Q. Lan, Nonexistence of the reversed flow solutions of the Falkner-Skan equations, Nonlinear Anal., 74 (2011), 5327-5339.
doi: dx.doi.org/10.1016/j.na.2011.05.017. |
[27] |
G. C. Yang and K. Q. Lan, The velocity and shear stress functions of the Falkner-Skan equation arising in boundary layer theory, J. Math. Anal. Appl., 328 (2007), 1297-1308.
doi: dx.doi.org/10.1016/j.jmaa.2006.06.042. |
[28] |
G. C. Yang, Existence of solutions of laminar boundary layer equations with decelerating external flows, Nonlinear Anal., 72 (2010), 2063-2075.
doi: dx.doi.org/10.1016/j.na.2009.10.006. |
[29] |
G. C. Yang, Existence of solutions to the third-order nonlinear differential equations arising in boundary layer theory, Appl. Math. Lett., 16 (2003), 827-832.
doi: dx.doi.org/10.1016/S0893-9659(03)90003-6. |
[30] |
G. C. Yang, A note on $f'''+ff''+\lambda (1-f^{'2})=0$ with $\lambda\in (-\frac{1}{2},0)$ arising in boundary layer theory, Appl. Math. Lett., 17 (2004), 1261-1265.
doi: dx.doi.org/10.1016/j.aml.2003.12.005. |
[31] |
G. C. Yang, L. L. Shi and K. Q. Lan, Properties of positive solutions of the Falkner-Skan equation arising in boundary layer theory, Integral Methods in Science and Engineering, 277-283, Birkhuser Boston, Boston, MA, 2008. |
show all references
References:
[1] |
R. P. Agarwal and D. O'Regan, Singular integral equations arising in Homann flow, Dyn. Cont. Discrete Impuls. Syst. Ser. B Appl. Algorithms, 9 (2002), 481-488. |
[2] |
D. J. Acheson, "Elementary Fluid Dynamics," Oxford Univ. Press, New York, 1990. |
[3] |
B. Brighi and J. D. Hoernel, On the concave and convex solutions of a mixed convection boundary layer approximation in a porous medium, Appl. Math. Lett., 19 (2006), 69-74.
doi: dx.doi.org/10.1016/j.aml.2005.02.038. |
[4] |
S. N. Brown and K. Stewartson, On the reversed flow solutions of the Falkner-Skan equations, Mathematika, 13 (1966), 1-6.
doi: dx.doi.org/10.1112/S0025579300004125. |
[5] |
W. A. Coppel, On a differential equation of boundary-layer theory, Phil. Trans. Roy. Soc. London, Ser. A, 253 (1960), 101-136.
doi: dx.doi.org/10.1112/S0025579300005052. |
[6] |
A. H. Craven and L. A. Peletier, On the uniqueness of solutions of the Falkner-Skan equation, Mathematika, 19 (1972), 129-133.
doi: dx.doi.org/10.1112/S0025579300005052. |
[7] |
I. G. Currie, "Fundamental Mechanics of Fluids," 3rd, Marcel Dekker, New York, 2003. |
[8] |
K. Deimling, "Nonlinear Functional Analysis," Springer-Verlag, Berlin, 1985.
doi: dx.doi.org/10.1007/978-3-662-00547-7. |
[9] |
M. Guedda and Z. Hammouch, On similarity and pseudo-similarity solutions of Falkner-Skan boundary layers, Fluid Dynam. Res., 38 (2006), 211-223.
doi: dx.doi.org/10.1016/j.fluiddyn.2005.11.001. |
[10] |
M. Guedda, Multiple solutions of mixed convenction boundary-layer approximations in a porous medium, Appl. Math. Lett., 19 (2006), 63-68.
doi: dx.doi.org/10.1016/j.aml.2005.02.037. |
[11] |
P. Hartman, "Ordinary Differential Equations," Classics in Applied Mathematics, 38, the Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. |
[12] |
P. Hartman, On the existence of similar solutions of some boundary layer problems, SIAM J. Math. Anal., 3 (1972), 120-147.
doi: dx.doi.org/10.1137/0503014. |
[13] |
S. P. Hastings, An existence theorem for a class of nonlinear boundary value problems including that of Falkner and Skan, J. Differential Equations, 9 (1971), 580-590.
doi: dx.doi.org/10.1016/0022-0396(71)90025-8. |
[14] |
S. P. Hastings, Reversed flow solutions of the Falkner-Skan equations, SIAM J. Appl. Math., 22 (1972), 329-334.
doi: dx.doi.org/10.1137/0122031. |
[15] |
K. Q. Lan and G. C. Yang, Positive solutions of the Falker-Skan equation arising in boundary layer theory, Canad. Math. Bull., 51 (2008), 386-398.
doi: dx.doi.org/10.4153/CMB-2008-039-7. |
[16] |
P. A. Libby and T. M. Liu, Further solutions of the Falkner-Skan equation, AIAA J., 5 (1967), 1040-1042.
doi: dx.doi.org/10.2514/3.4130. |
[17] |
T. Y. Na, "Computational Methods in Engineering Boundary Value Problems," Academic Press, 1979. |
[18] |
H. Schlichting and K. Gersten, "Boundary-Layer Theory," 8ed, Springer-Verlag, Berlin, 2000. |
[19] |
K. Schrader, A generalization of the Helly selection theorem, Bull. Amer. Math. Soc., 78 (1972), 415-419.
doi: dx.doi.org/10.1090/S0002-9904-1972-12923-9. |
[20] |
C. Sparrow and H. P. F. Swinnerton-Dyer, The Falkner-Skan equation. II. Dynamics and the bifurcations of $P$- and $Q$-orbits, J. Differential Equations, 183 (2002), 1-55.
doi: dx.doi.org/10.1006/jdeq.2001.4100. |
[21] |
K. Stewartson, Further solutions of the Falkner-Skan equation, Proc. Camb. Phil. Soc., 50 (1954), 454-465.
doi: dx.doi.org/10.1017/S030500410002956X. |
[22] |
H. P. F. Swinnerton-Dyer and C. T. Sparrow, The Falkner-Skan equation. I. The creation of strange invariant sets, J. Differential Equations, 119 (1995), 336-394.
doi: dx.doi.org/10.1006/jdeq.1995.1094. |
[23] |
K. K. Tam, A note on the existence of a solution of the Falkner-Skan equation, Canad. Math. Bull., 13 (1970), 125-127.
doi: dx.doi.org/10.4153/CMB-1970-026-8. |
[24] |
J. Wang, W. Gao and Z. Zhang, Singular nonlinear boundary value problems arising in boundary layer theory, J. Math. Anal. Appl., 233 (1999), 246-256.
doi: dx.doi.org/10.1006/jmaa.1999.6290. |
[25] |
H. Weyl, On the differential equations of the simplest boundary-layer problems, Ann. Math., 43 (1942), 381-407.
doi: dx.doi.org/10.2307/1968875. |
[26] |
G. C. Yang and K. Q. Lan, Nonexistence of the reversed flow solutions of the Falkner-Skan equations, Nonlinear Anal., 74 (2011), 5327-5339.
doi: dx.doi.org/10.1016/j.na.2011.05.017. |
[27] |
G. C. Yang and K. Q. Lan, The velocity and shear stress functions of the Falkner-Skan equation arising in boundary layer theory, J. Math. Anal. Appl., 328 (2007), 1297-1308.
doi: dx.doi.org/10.1016/j.jmaa.2006.06.042. |
[28] |
G. C. Yang, Existence of solutions of laminar boundary layer equations with decelerating external flows, Nonlinear Anal., 72 (2010), 2063-2075.
doi: dx.doi.org/10.1016/j.na.2009.10.006. |
[29] |
G. C. Yang, Existence of solutions to the third-order nonlinear differential equations arising in boundary layer theory, Appl. Math. Lett., 16 (2003), 827-832.
doi: dx.doi.org/10.1016/S0893-9659(03)90003-6. |
[30] |
G. C. Yang, A note on $f'''+ff''+\lambda (1-f^{'2})=0$ with $\lambda\in (-\frac{1}{2},0)$ arising in boundary layer theory, Appl. Math. Lett., 17 (2004), 1261-1265.
doi: dx.doi.org/10.1016/j.aml.2003.12.005. |
[31] |
G. C. Yang, L. L. Shi and K. Q. Lan, Properties of positive solutions of the Falkner-Skan equation arising in boundary layer theory, Integral Methods in Science and Engineering, 277-283, Birkhuser Boston, Boston, MA, 2008. |
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