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A new approach to energy theory in the stability of fluid motion

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Bibliography

  • Adams, R. A., (1975), Sobolev Spaces, Academic Press, N.Y., S. Francisco, London.

    Google Scholar 

  • Agmon, S., Douglis, A., & L. Nirenberg, (1959), Comm. Pure Appl. Math., 12, 623.

    Google Scholar 

  • Agrawal, S. C., (1969), J. Phys. Soc. Japan, 26, 561.

    Google Scholar 

  • Aris, R., (1975), The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts, Vol. II, Oxford.

  • Babovsky, H., & M. Padula (forthcoming), New contributions to nonlinear stability of a discrete velocity model.

  • Banerjee, M. B., Gupta, J. R., Shandil, R. G., & S. K. Sood, (1985), J. Math. Anal. and Appl., 108, 216.

    Google Scholar 

  • Bogovskii, M. E., (1979), Soviet Math. Dokl., 20, 1094.

    Google Scholar 

  • Braginskii, S. I., (1957), J. Exptl. Theoret. Phys., 33, 459.

    Google Scholar 

  • Busse, F. H., (1975), J. Fluid Mech., 71, 193.

    Google Scholar 

  • Cattabriga, L., (1961), Rend. Mat. Sem. Univ. Padova, 31, 308.

    Google Scholar 

  • Chandrasekhar, S., (1981), Hydrodynamic and Hydromagnetic Stability, Dover Publ. Inc., N.Y.

    Google Scholar 

  • Chandrasekhar, S., & D. D. Elbert, (1955), Proc. Roy. Soc. London (A), 231, 198.

    Google Scholar 

  • Coscia, V., & M. Padula, (1989), Hydrosoft, 2 (4), 182.

    Google Scholar 

  • Coscia, V., & M. Padula, (1990), Geophys. Astrophys. Fluid Dyn., in press.

  • Cowling, T. G., (1957), Magnetohydrodynamics, Intersci. N.Y.

    Google Scholar 

  • Donnelly, R. J., & M. Ozima, (1960), Phys. Rev. Letters, 4, 497.

    Google Scholar 

  • Drazin, P. G., & W. H. Reid, (1982), Hydrodynamic Stability, Cambridge Monographs on Mech. and Appl. Math., Cambridge Univ. Press.

  • Ebin. D., & M. C. Shen, (1987), J. Math. Anal. and Appl., 125, 81.

    Google Scholar 

  • Ebin, D., & M. C. Shen, (1988), Ann. Mat. Pura Appl., 50, 39.

    Google Scholar 

  • Fife, P., (1979), Mathematical Aspects of Reacting and Diffusing Systems, Springer Lecture Notes in Mathematics, 28.

  • Galdi, G. P., (1975), Arch. Rational Mech. Anal., 59, 1.

    Google Scholar 

  • Galdi, G. P., (1979), Ricerche Mat., 27, 387.

    Google Scholar 

  • Galdi, G. P., (1985), Arch. Rational Mech. Anal., 87, 167.

    Google Scholar 

  • Galdi, G. P., (1989), Proc. Fourth Intl Conf. on Computational Methods and Experimental Measurements, Eds. G. M. Carlomagno & C. A. Brebbia, Springer-Verlag Berlin Heidelberg N. Y. London Paris Tokyo, 183.

    Google Scholar 

  • Galdi, G. P., An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Springer Tracts in Natural Philosophy, forthcoming.

  • Galdi, G. P., & I. Herron, (1985), Quart. Appl. Math., 52, 159.

    Google Scholar 

  • Galdi, G. P., & M. Padula, (1990), Proc. Third Workshop on Math. Aspects of Fluids and Plasma Dynamics, G. Toscani, V. Boffi & S. Ronero Eds., Springer Lecture Notes in Math., in Press.

  • Galdi, G. P., & S. Rionero, (1985), Weighted Energy Methods in Fluid Dynamics and Elasticity, Springer Lecture Notes in Mathematics, 1134

  • Galdi, G. P., & B. Straughan, (1985a), Arch. Rational Mech. Anal., 89, 211.

    Google Scholar 

  • Galdi, G. P., & B. Straughan, (1985b), Proc. Roy. Soc. London (A), 402, 257.

    Google Scholar 

  • Galdi, G. P., Payne, L. H., Proctor, M. R. E., & B. Straughan, (1987), Proc. Roy. Soc. London (A), 414, 83.

    Google Scholar 

  • Galdi, G. P., Padula, M., & K. R. Rajagopal, (1990), Arch. Rational Mech. Anal., 109, 173.

    Google Scholar 

  • Gavalas, G. R., (1968), Nonlinear Differential Equations of Chemically Reacting Systems, Springer-Verlag, N.Y.

    Google Scholar 

  • Glazman, I. M., (1965), Direct Methods of Qualitative Spectral Analysis of Singular Differential Operators, Israel Progr. for Sci. Transl., Jerusalem.

  • Goody, R. M., (1956), J. Fluid Mech., 1, 424.

    Google Scholar 

  • Guckenheimer, J., & E. Knobloch, (1983), Geophys. Astrophys. Fluid Dyn., 23, 247.

    Google Scholar 

  • Henry, D., (1981), Geometric Theory of Semilinear Parabolic Equations, Springer Lecture Notes in Mathematics, 840.

  • Hetrick, D. L., (1971), Dynamics of Nuclear Reactors, Univ. Chicago Press, Chicago.

    Google Scholar 

  • Heywood, J. G., (1980), Indiana Univ. Math. J., 21, 639.

    Google Scholar 

  • Kadish, A., (1976), Phys. Fluids, 19, 141.

    Google Scholar 

  • Kappraff, J., Grossmann, W., & M. Dress, (1981), J. Plasma Physics, 25, 111.

    Google Scholar 

  • Kato, T., (1966), Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, Heidelberg, N. Y.

    Google Scholar 

  • Kaufman, A., (1960), La Théorie des Gas Neutres et Ionizés, Herman et Cie., Paris.

    Google Scholar 

  • Kloeden, P., & R. Wells, (1983), Proc. Roy. Soc. London (A), 390, 293.

    Google Scholar 

  • Kolyshkin, A. A., (1987), Magnitnaya Gidrodinamika, 1, 67 (in Russian).

    Google Scholar 

  • Krishnamurthy, R., (1967), Ph. D. Thesis, Dept. Geophysics and Planetary Science, UCLA.

  • Jorné, J., & S. Carmi, (1977), Math. Biosciences, 37, 51.

    Google Scholar 

  • Joseph, D. D., (1965), Arch. Rational Mech. Anal, 20, 59.

    Google Scholar 

  • Joseph, D. D., (1966), Arch. Rational Mech. Anal., 22, 163.

    Google Scholar 

  • Joseph, D. D., (1970), Arch. Rational Mech. Anal., 36, 285.

    Google Scholar 

  • Joseph, D. D., (1976), Stability of Fluid Motions, Springer Tracts in Natural Philosophy, Volumes I and II.

  • Ladyzhenskaya, O. A., (1969), The Mathematical Theory of Viscous Incompressible Flow, Gordon & Breach Sci. Publ., N. Y., London, Paris.

    Google Scholar 

  • Langlois, W. E., (1981), Physico-Chem. Hydrodyn., 2, 245.

    Google Scholar 

  • Liubimov, H. A., (1962), J. Mech. Appl. Math. (PMM), 26, 789.

    Google Scholar 

  • Maiellaro, M., & L. Palese, (1984), Int. J. Engng. Sci., 22, 411.

    Google Scholar 

  • Zaremonti, P., (1984), Rend. Sem. Mat. Univ. Padova, 71, 35.

    Google Scholar 

  • Masuda, K., (1975), J. Math. Soc. Japan, 27, 294.

    Google Scholar 

  • Mattei, G., (1970), Ann. Mat. Pura Appl., 80, 1.

    Google Scholar 

  • Miklavčic, M., (1985), Pacific J. of Math., 118, 199.

    Google Scholar 

  • Mulone, G., & S. Rionero, (1988), J. Math. Anal. and Appl., in Press.

  • Nečas, J., (1967), Les Méthodes Directes en Théorie des Equations Elliptiques, Masson et Cie., Prague.

    Google Scholar 

  • Nicolis, G., & I. Prigogine, (1977), Self Organization in Nonequilibrium Systems, John Wiley & Sons Inc., N.Y., London, Sydney, Toronto.

    Google Scholar 

  • Orr, W. McF., (1907), Proc. Roy. Irish Acad. (A), 27, 69.

    Google Scholar 

  • Padula, M., (1984), Proc. Roy. Soc. Edinburgh, 96, 55.

    Google Scholar 

  • Padula, M., (1986), Boll. U.M.I., 5B, 581.

    Google Scholar 

  • Padula, M., (1988a), Energy Instability Methods: an Application to Burgers Equation, Proc. Meeting Energy Stability and Convection, Galdi, G. P., & B. Straughan Eds., Pitman Research Notes in Mathematics, 168.

  • Padula, M., (1988b), Compressible Convection: Nonlinear Results, Proc. Intl. Conf. on “Mathematical Modelling in Science and Technology”, Madras, India.

  • Padula, M., (1990), Proc. Conf. on Waves and Stability in Continuous Media, S. Rionero Ed., World Sci. Publ., in Press.

  • Peckover, R. S., & N. O. Weiss, (1972), Comp. Phys. Comm., 4, 339.

    Google Scholar 

  • Proctor, M. R. E., & D. J. Galloway, (1979), J. Fluid Mech., 90, 273.

    Google Scholar 

  • Proctor, M. R. E., & N. O. Weiss, (1982), Rep. Progr. Phys., 45, 1317.

    Google Scholar 

  • Rai, L., (1968), J. Phys., 46, 2533.

    Google Scholar 

  • Rayleigh, Lord, (1916), Proc. Roy. Soc. London (A), 93, 148.

    Google Scholar 

  • Reynolds, O., (1895), Phil. Trans. Roy. Soc. London (A), 186, 123.

    Google Scholar 

  • Rionero, S., (1968), Ann. Mat. Pura Appl., 78, 339.

    Google Scholar 

  • Rionero, S., (1971), Ricerche Mat., 20, 285.

    Google Scholar 

  • Rionero, S., (1988), On the Choice of the Liapunov Functional in the Stability of Fluid Motions, Proc. Meeting “Energy Stability and Convection”, Galdi, G. P., & B. Straughan Eds., Pitman Research Notes in Mathematics, 168.

  • Rionero, S., & G. Mulone, (1988), Arch. Rational Mech. Anal., 103, 347.

    Google Scholar 

  • Roberts, P. H., (1964), Proc. Camb. Phil. Soc., 60, 635.

    Google Scholar 

  • Roberts, P. H., (1967a), An Introduction to Magnetohydrodynamics, Longmans: London.

    Google Scholar 

  • Roberts, P. H., (1967b), J. Fluid Mech., 30, 33.

    Google Scholar 

  • Roberts, P. H., & K. Stewartson, (1974), Phil. Trans. Roy. Soc. Lond. (A), 277, 287.

    Google Scholar 

  • Rossby, H. T., (1969), J. Fluid Mech., 36, 306.

    Google Scholar 

  • Rudraiah, N., (1981), Publ. Astron. Soc. Japan, 33, 721.

    Google Scholar 

  • Rudraiah, N., Kumudini, V., & W. Unno, (1985), Publ. Astron. Soc. Japan, 37, 183.

    Google Scholar 

  • Rudraiah, N., & M. Sekhar, (1988), Effect of Non-uniform Temperature Gradient on Convection in Magnetic Fluids, preprint of the Central College, Bangalore.

  • Sattinger, D. S., (1973), Topics in Stability and Bifurcation Theory, Springer Lect. Notes in Math., 309, Berlin, Heidelberg, N. Y.

  • Segel, L. A., & J. L. Jackson, (1972), J. Theor. Biol., 37, 545

    Google Scholar 

  • Serrin, J., (1959), Arch. Rational Mech. Anal., 3, 1.

    Google Scholar 

  • Solonnikov, V. A., & Y. Scadilov, (1973), Trudy Mat. Inst. Steklov, 125, 186.

    Google Scholar 

  • Solonnikov, V. A., & D. Pileckas, (1986), J. Soviet Mat., 34, 2101.

    Google Scholar 

  • Sparrow, E. M., Goldstein, R. J., & V. K. Jonsson, (1964), J. Fluid Mech., 18, 513.

    Google Scholar 

  • Spitzer, J., (1964), Physics of Fully Ionized Gases, Intersci. Tracts on Phys. and Astron.

  • Straughan, B., (1982), Portugaliae Matematica, 41, 251.

    Google Scholar 

  • Straughan, B., (1988), Convection in a variable gravity field, preprint of Univ. Glasgow

  • Synge, J. L., (1933), Trans. of the Royal Society of Canada, 27, 1.

    Google Scholar 

  • Thompson, W. B., (1951), Phil. Mag., 42, 1417.

    Google Scholar 

  • Veronis, G., (1959), J. Fluid Mech., 5, 401.

    Google Scholar 

  • Weiss, N. O., (1981), J. Fluid Mech., 108, 247.

    Google Scholar 

  • Youdovich, V. I., (1967), Math. USSR Sbornik, 3, 519.

    Google Scholar 

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Communicated by K. R. Rajagopal

Dedicated to Clifford Truesdell, on his 70th birthday

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Galdi, G.P., Padula, M. A new approach to energy theory in the stability of fluid motion. Arch. Rational Mech. Anal. 110, 187–286 (1990). https://doi.org/10.1007/BF00375129

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