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References
Chandrasekhar, S., Hydrodynamic and Hydromagnetic Stability. Oxford, 1961.
Mihaljan, J., A rigorous exposition of the Boussinesq approximations applicable to a thin layer of fluid. Astrophy J. 136, 1126–1133 (1962).
Joseph, D. D., Nonlinear stability of the Boussinesq equations by the method of energy. Arch. Rational Mech. Anal. (3) 22, 163–184 (1966).
Segel, L., Nonlinear Hydrodynamic Stability Theory and its Application to Thermal Convection and Curved Flow in Non-Equilibrium Thermodynamics: Variational Techniques and Stability. Univ. Chicago Press, 1966.
Görtler, H., & W. Velte, Recent mathematical treatments of laminar flow and transition problems. Phys. Fluids 10, 33 (1967).
Ukhovskii, M. R., & V. T. Judovich, On the equation of steady state convection. Prik. Math. Mekl. 27, 353–370 (1963).
Sani, R., On the non-existence of subcritical instability in fluid layers heated from below. J. Fluid Mech. 20, 315–319 (1964).
Howard, L. N., Heat transport by turbulent convection. J. Fluid Mech. 17, 405–432 (1963).
Velte, W., Stabilität und Verzweigung Stationärer Lösungen der Navier-Stokesschen Gleichungen. Arch. Rational Mech. Anal. 22, 1–14 (1966).
Rabinowitz, P., Existence and nonuniqueness of rectangular solutions of the Bénard problem. Arch. Rational Mech. Anal. 29, 32–57 (1968).
Judovich, V. I., The appearance of convection. Prikl. Matem, i Mekh 30, 1000–1005 (1966), English Translation, PMM 30, 1193–1199 (1967).
Kirchgässner, K., Habilitationsschrift, Universität Freiburg (1966); see also Reference 5.
Schlüter, A., D. Lortz, & F. Busse, On the stability of steady finite amplitude cellular convection. J. Fluid Mech. 23, 129–144 (1965).
Gorkov, L., Steady convection in a plane liquid layer near the critical heat transfer point. Sov. Phys. JETP 6, 311 (1957).
Malkus, W., & G. Veronis, Finite amplitude cellular convection. J. Fluid Mech. 4, 225–260 (1958).
Busse, F., Dissertation, Munich (1962). Cf. also The stability of finite amplitude cellular convection and its relation to an extremum principle. J. Fluid Mech. 30, 625–651 (1967).
Krishnamurti, R., Finite amplitude convection with changing mean temperature. J. Fluid Mech. 33, 445–465 (1968).
Vorovich, I. I., & V. I. Judovich, Steady motion of a viscous incompressible fluid. Matem. Sb. 53, 393–428 (1961).
Ladyzhenskaya, O. A., The Mathematical Theory of Viscous Incompressible Flow. New York: Gordon and Breach 1963.
Fujita, H., On the existence and regularity of the steady-state solutions of the Navier-Stokes equations. J. Fac. Sci. Univ. Tokyo Sec. I, 9, 59–102 (1961).
Edwards, J. E., On the existence of solutions of the steady-state Navier-Stokes equations for a class of non-smooth boundary data. Technical Report, Lockheed Missiles and Space Company, June, 1963.
Agmon, S., A. Douglis, & L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II. Comm. Pure Appl. Math. 17, 35–42 (1964).
Joseph, D., R. Goldstein, & D. Graham, Subcritical instability and exchange of stability in a horizontal fluid layer. Phys. Fluids 11, 903–904 (1968).
Lortz, D., Dissertation, Munich (1961).
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Communicated by H. Görtler
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Fife, P.C., Joseph, D.D. Existence of convective solutions of the generalized Bénard problem which are analytic in their norm. Arch. Rational Mech. Anal. 33, 116–138 (1969). https://doi.org/10.1007/BF00247756
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DOI: https://doi.org/10.1007/BF00247756