Abstract
Our aim in this article is to study the existence and regularity of solutions of a quasilinear elliptic-hyperbolic equation. This equation appears in the design of blade cascade profiles. This leads to an inverse problem for designing two-dimensional channels with prescribed velocity distributions along channel walls. The governing equation is obtained by transformation of the physical domain to the plane defined by the streamlines and the potential lines of fluid. We establish an existence and regularity result of solutions for a more general framework which includes our physical problem as a specific example.
Similar content being viewed by others
References
Boccardo, L., Murat, F., and Puel, J. P.: Existence de solutions faibles pour des équations elliptiques quasilinéaires à croissance quadradique, in H. Brézis and J. L. Lions (eds.), Nonlinear Part. Diff. Equ. and Their Applications, Collège de France, séminaire Vol. IV Pitman, London, 1983, pp. 19–73; see also Ann. della Scuola Norm. Sup. Pisa XI (1984), 213–235.
Browder, F. E.: Existence theorems for nonlinear partial differential equations, in S. S. Chern and S. Smale (eds.) Symposia in Pure Mathematics, Vol. XVI, Amer. Math. Soc., Providence, R.I., 1970, pp. 1–60.
Castellant, J. P.: Introduction géométrique à la mécanique des fluides, Etude SNECMA, Février 1986.
Giqueaux, M.: Mécanique des fluides théorique, Librairie Polytechnique Béranger, Paris and Liège, 1957.
Hess, P.: On a second order nonlinear elliptic boundary value problem, in L. Cesari, R. Kannan and H. F. Weinberger (eds.) Nonlinear Analysis: A Collection of Papers in Honor of Erich. H. Rothe, Academic Press, New York, 1978, pp. 99–107
Hawthorne, W. R., (ed.): High Speed Aerodynamics and Jet Propulsion: Aerodynamics of Turbines and Compressors, Vol. X, Princeton University Press, 1964.
Ladyzens'kaja, O. A. and Uraltceva, N. N.: Linear and Quasilinear Elliptic Equations, Academic Press, New York, London, 1968.
Lions, J. L.: Quelques méthodes de résolution des problèmes aux limites nonlinéaires, Dunod, Paris, 1969.
Michaux, B.: Méthode inverse de détermination de canaux courbes transsoniques, Rapport SNECMA, Octobre 1986.
Michaux, B.: Méthodes inverses pour la détermination des profils d'aubes transsoniques des turbomachines, Thése, Univ. de Paris-Sud, 1988.
Michaux, B., Rakotoson, J. M., and Shen, J.: On the approximation of a quasilinear mixed problem. (Inst. Appl. Math. Anl. Scient. Comp. Preprint # 8808 (1988), Indiana University.) To appear in Modélisation Mathématique et Analyse Numérique.
Moser, J.: A new proof of De Gorgi's theorem concerning the regularity problem for elliptic differential equations, Commun. Pure Appl. Math. 13 (1960), 457–468.
Mossino, J. and Rakotoson, J. M.: Isoperimetric inequalities in parabolic equations, Ann. della Scuola Norm. Sup. Pisa (4) XII (1986), 51–73.
Mossino, J. and Temam, R.: Directional derivative of the increasing rearrangement mapping and application to a queer differential equation in plasma physics, Duke Math. J. 41 (1981), 475–495.
Rakotoson, J. M.: Quelques propriétés du réarrangements relatif, C.R. Acad. Sci. Paris serie I, 302, No. 15 (1986).
Rakotoson, J. M.: Réarrangement relatif dans les équations quasilinéaires avec un second membre distribution: Application à un résultat d'existence et de régularité, J. Diff. Equ. 66, No. 6 (1987).
Rakotoson, J. M.: Existence of bounded solutions of some degenerate quasilinear elliptic equations, Commun. Part. Diff. Equ., 12 (6), (1987), 633–676.
Rakotoson, J. M.: On some degenerate and nondegenerate quasilinear elliptic systems with nonhomogeneous Dirichlet boundary condition, To appear in Nonlinear Analysis Th. Meth. Appl.
Rakotoson, J. M. and Temam, R.: Relative rearrangement in quasilinear variational inequalities, Preprint d'Orsay, France (1987), Ind. Univ. Math. J. 36 (1987).
Stanitz, J. D.: Design of two-dimensional channels with prescribed velocity distributions along channels walls, NACA 2593/2595 (1959).
Stanitz, J. D. and Sheldrake, L.T.: Application of design method to high-solidity cascades and tests of an impulse cascade with 90° of turning, NACA TN 2652 (1962).
Trudinger, N. S.: On Harnack inequalities and their application to quasilinear elliptic equations, Commun. Pure Appl. Math. (1967).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Michaux, B., Rakotoson, J.M. & Shen, J. On the existence and regularity of solutions of a quasilinear mixed equation of Leray-Lions type. Acta Appl Math 12, 287–316 (1988). https://doi.org/10.1007/BF00046884
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00046884