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Article

Ladopoulos, E. G. ; Zisis, V. A.
Existence and uniqueness for non-linear singular integral equations used in fluid mechanics. (English). Applications of Mathematics, vol. 42 (1997), issue 5, pp. 345-367
MSC: 45E05, 65L10, 65R20, 76M99 | MR 1467554 | Zbl 0906.76076
Keywords:
non-linear singular integral equations; existence and uniqueness theorems; Banach spaces; Hölder conditions; fluid mechanics
Summary:

References:
[1] E. G. Ladopoulos: On the numerical solution of the finite-part singular integral equations of the first and the second kind used in fracture mechanics. Comp. Meth. Appl. Mech. Engng 65 (1987), 253–266. MR 0919245
[2] E. G. Ladopoulos: On a new integration rule with the Gegenbauer polynomials for singular integral equations, used in the theory of elasticity. Ing. Archiv 58 (1988), 35–46. Zbl 0627.73018
[3] E. G. Ladopoulos: On the numerical evaluation of the general type of finite-part singular integrals and integral equations used in fracture mechanics. J. Engrg. Fract. Mech. 31 (1988), 315–337.
[4] E. G. Ladopoulos: The general type of finite-part singular integrals and integral equations with logarithmic singularities used in fracture mechanics. Acta Mech. 75 (1988), 275–285. Zbl 0667.73072
[5] E. G. Ladopoulos: On the solution of the two-dimensional problem of a plane crack of arbitrary shape in an anisotropic material. J. Engrg Fract. Mech. 28 (1987), 187–195.
[6] E. G. Ladopoulos: On the numerical evaluation of the singular integral equations used in two and three-dimensional plasticity problems. Mech. Res. Commun. 14 (1987), 263–274. Zbl 0635.73044
[7] E. G. Ladopoulos: Singular integral representation of three-dimensional plasticity fracture problem. Theor. Appl. Fract. Mech. 8 (1987), 205–211.
[8] E. G. Ladopoulos: On the numerical solution of the multidimensional singular integrals and integral equations used in the theory of linear viscoelasticity. Internat. J. Math. Math. Scien. 11 (1988), 561–574. MR 0947288 | Zbl 0665.65097
[9] E. G. Ladopoulos: Relativistic elastic stress analysis for moving frames. Rev. Roum. Sci. Tech., Méc. Appl. 36 (1991), 195–209. MR 1171626
[10] J. Andrews and J. M. Ball: Asymptotic behaviour and changes of phase in one-dimensional nonlinear viscoelasticity. J. Diff. Eqns 44 (1982), 306–341. MR 0657784
[11] S. S. Antman: Ordinary differential equations of nonlinear elasticity I: Foundations of the theories of nonlinearly elastic rods and shells. Arch. Ration. Mech. Anal. 61 (1976), 307–351. MR 0418580
[12] S. S. Antman: Ordinary differential equations of nonlinear elasticity II: Existence and regularity for conservative boundary value problems. Arch. Ration. Mech. Anal. 61 (1976), 352–393.
[13] S. S. Antman, E. R. Carbone: Shear and necking instabilities in nonlinear elasticity. J. Elasticity 7 (1977), 125–151. MR 0451990
[14] J. M. Ball: Convexity conditions and existence theorems in nonlinear elasticity. Arch. Ration. Mech. Anal. 63 (1977), 337-403. MR 0475169 | Zbl 0368.73040
[15] J. M. Ball: Discontinuous equilibrium solutions and cavitation in nonlinear elasticity. Phil. Trans. R. Soc. Lond. A 306 (1982), 557–611. MR 0703623 | Zbl 0513.73020
[16] J. M. Ball: Remarques sur l’existence et la régularité des solutions d’elastostatique nonlinéaire, in: Recent Contributions to Nonlinear Partial Differential Equations. Pitman, Boston, 1981, pp. 50–62. MR 0639745
[17] H. Brezis: Equations et inéquations non lineaires dans les éspaces vectoriels en dualite. Ann. Inst. Fourier 18 (1968), 115–175. MR 0270222 | Zbl 0169.18602
[18] P. G. Ciarlet, P. Destuynder: A justification of a nonlinear model in plate theory. Comp. Meth. Appl. Mech. Engng 17 (1979), 227–258. MR 0533827
[19] P. G. Ciarlet, J. Nečas: Injectivité presque partout, autocontact, et noninterpénétrabilité en élasticité non linéaire tridimensionnelle. C. R. Akad. Sci. Paris, Sér I 301 (1985), 621–624. MR 0816644
[20] P. G. Ciarlet, J. Nečas: Injectivity and self-contact in non-linear elasticity. Arch. Ration. Mech. Anal. 97 (1987), 171–188. MR 0862546
[21] C. M. Dafermos: The mixed initial-boundary value problem for the equations of nonlinear one dimensional viscoelasticity. J. Diff. Eqns 6 (1969), 71–86. MR 0241831 | Zbl 0218.73054
[22] C. M. Dafermos: Development of singularities in the motion of materials with fading memory. Arch. Ration. Mech. Anal. 91 (1985), 193–205. MR 0806001
[23] C. M. Dafermos, L. Hsiao: Development of singularities in solutions of the equations of nonlinear thermoelasticity. Q. Appl. Math. 44 (1986), 463–474. MR 0860899
[24] J. E. Dendy: Galerkin’s method for some highly nonlinear problems. SIAM J. Num. Anal. 14 (1977), 327–347. MR 0433914 | Zbl 0365.65065
[25] Guo Zhong-Heng: The unified theory of variational principles in nonlinear elasticity. Arch. Mech. 32 (1980), 577–596. MR 0619303
[26] H. Hattori: Breakdown of smooth solutions in dissipative nonlinear hyperbolic equations. Q. Appl. Math. 40 (1982), 113–127. MR 0666668 | Zbl 0505.76008
[27] D. Hoff, J. Smoller: Solutions in the large for certain nonlinear parabolic systems. Anal. Non Lin. 2 (1985), 213–235. MR 0797271
[28] W. J. Hrusa: A nonlinear functional differential equation in Banach space with applications to materials with fading memory. Arch. Ration. Mech. Anal. 84 (1983), 99–137. MR 0713121 | Zbl 0544.73056
[29] R. C. MacCamy: Nonlinear Volterra equations on a Hilbert space. J. Diff. Eqns 16 (1974), 373–393. MR 0377605 | Zbl 0263.45010
[30] R. C. MacCamy: Stability theorems for a class of functional differential equations. SIAM J. Appl. Math. 30 (1976), 557–576. MR 0404818 | Zbl 0346.34059
[31] R. C. MacCamy: A model for non-dimensional, nonlinear viscoelasticity. Q. Appl. Math. 35 (1977), 21–33. MR 0478939
[32] B. Neta: Finite element approximation of a nonlinear parabolic problem. Comput. Math Appl. 4 (1987), 247–255. MR 0518696
[33] B. Neta: Numerical solution of a nonlinear integro-differential equation. J. Math. Anal. Appl. 89 (1982), 598–611. MR 0677747 | Zbl 0488.65074
[34] R. W. Ogden: Principal stress and strain trajectories in nonlinear elastostatics. Q. Appl. Math. 44 (1986), 255–264. MR 0856179 | Zbl 0608.73022
[35] R. L. Pego: Phase transitions in one-dimensional nonlinear viscoelasticity: admissibility and stability. Arch. Ration. Mech. Anal. 97 (1987), 353–394. MR 0865845 | Zbl 0656.73023
[36] M. Slemrod: Global existence, uniqueness and asymptotic stability of classical smooth solutions in one-dimensional, nonlinear thermoelasticity. Arch. Ration. Mech. Anal. 76 (1981), 97–133. MR 0629700 | Zbl 0481.73009
[37] O. J. Staffans: On a nonlinear hyperbolic Volterra equation. SIAM J. Math. Anal. 11 (1980), 793–812. MR 0586908 | Zbl 0464.45010
[38] J. Schauder: Der Fixpunktsatz in Funktionalräumen. Studia Math. 2 (1930), 171–180.
[39] I. Privalov: On a boundary problem in the theory of analytic functions. Math. Sb. 41 (1934), 519–526.
[40] S. Banach: Sur les opérations dans les ensembles abstraits et leurs applications aux équations intégrales. Fundam. Math. 3 (1922), 133–181.
[41] V. Cacciopoli: Un teorema generale sull’asistenza di elementi uniti in una transformazione functionale. Rend. Accad. Lincei 2 (1930).
[42] M. I. Zhykovskiy: Calculation of the Flow in Lattices of Profiles of Turbomachines. Mashgiz, Moscow, 1960. (Russian)
[43] N. I. Muskhelishvili: Some Basic Problems of the Mathematical Theory of Elasticity. Noordhoff, Groningen, Netherlands, 1953. MR 0058417 | Zbl 0052.41402
[44] N. I. Muskhelishvili: Singular Integral Equations. Noordhoff, Groningen, Netherlands, 1972. MR 0355494
[45] V. V. Ivanov: The Theory of Approximate Methods and their Application to the Numerical Solution of Singular Integral Equations. Noordhoff, Leyden, Netherlands, 1976. MR 0405045 | Zbl 0346.65065
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