Skip to main content
Log in

On the Navier-Stokes initial value problem. I

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Bibliography

  1. Cattabriga, L., Su un problema al contorno relativo al sistema di equazioni di Stokes. Rendiconti Seminario Mat. Univ. Padova, 31, 1–33 (1961).

    Google Scholar 

  2. Golovkin, K. K., & B. A. Solonnikov, On the first boundary value problem for the non-stationary Navier-Stokes equation. Doklady Akad. Nauk USSR 140, 287–290 (1961).

    Google Scholar 

  3. Fujita, H., On the existence and regularity of the steady-state solutions of the Navier-Stokes equation. J. Fac. Sci., Univ. Tokyo, Sec. I 9, 59–102 (1961).

    Google Scholar 

  4. Fujita, H., Unique existence of solutions of the Navier-Stokes initial value problem, (an application of fractional powers of operators). Sûgaku (Iwanami) 14, 65–81 (1962).

    Google Scholar 

  5. Hopf, E., Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen. Math. Nachr. 4, 213–231 (1951).

    Google Scholar 

  6. Ito, S., The existence and the uniqueness of regular solution of non-stationary Navier-Stokes equation. J. Fac. Sci., Univ. Tokyo, Sec. I 9, 103–140 (1961).

    Google Scholar 

  7. Kato, T., Abstract evolution equation of parabolic type in Banach and Hilbert spaces. Nagoya Math. J. 19, 93–125 (1961).

    Google Scholar 

  8. Kato, T., Fractional powers of dissipative operators. J. Math. Soc. Japan 13, 246–274 (1961).

    Google Scholar 

  9. Kato, T., A generalization of the Heinz inequality. Proc. Japan Acad. 37, 305–308 (1961).

    Google Scholar 

  10. Kato, T., & H. Fujita, On the non-stationary Navier-Stokes system. Rendiconti Seminario Math. Univ. Padova 32, 243–260 (1962).

    Google Scholar 

  11. Kiselev, A. A., & O. A. Ladyzhenskaia, On existence and uniqueness of the solution of the non-stationary problem for any incompressible viscous fluid. Izv. Akad. Nauk. USSR, 21, 655–680 (1957).

    Google Scholar 

  12. Ladyzhenskaia, O. A., Solution “in the large” of the non-stationary boundary value problem for the Navier-Stokes system with two space variables. Comm. Pure Appl. Math. 12, 427–433 (1959).

    Google Scholar 

  13. Ladyzhenskaia, O. A., Mathematical Problems for Dynamics of Viscous Incompressible Fluids. Moscow 1961.

  14. Leray, J., Étude de diverses équations intégrales non linéaires et de quelques problèmes que pose l'hydrodynamique. J. Math. Pures Appl., Ser. IX 12 1–82 (1933).

    Google Scholar 

  15. Leray, J., Sur le mouvement d'un liquide visqueux emplissant l'espace. Acta Math. 63, 193–248 (1934).

    Google Scholar 

  16. Lions, J. L., Sur la régularité et l'unicité des solutions turbulentes des équations de Navier-Stokes. Rendiconti Seminario Mat. Univ. Padova 30, 16–23 (1960).

    Google Scholar 

  17. Lions, J. L., Sur les espaces d'interpolation; dualité. Math. Scand. 9, 147–177 (1961).

    Google Scholar 

  18. Lions, J. L., & G. Prodi, Un théorème d'existence et unicité dans les équations de Navier-Stokes en dimension 2. C.R. Acad. Sci. Paris 248, 3519–3521 (1959).

    Google Scholar 

  19. Odqvist, F. K. G., Über die Randwertaufgabe der Hydrodynamik zäher Flüssigkeiten. Math. Z. 32, 329–375 (1930).

    Google Scholar 

  20. Ohyama, T., Interior regularity of weak solutions of the time-dependent Navier Stokes equation. Proc. Japan Acad. 36, 273–277 (1960).

    Google Scholar 

  21. Oseen, C. W., Hydrodynamik. Leipzig 1927.

  22. Serrin, J., On the interior regularity of weak solutions of the Navier-Stokes equation. Arch. Rational Mech. Anal. 9, 187–195 (1962).

    Google Scholar 

  23. Sobolevskii, P. E., On non-stationary equations of hydrodynamics for viscous fluid. Doklady Akad. Nauk USSR 128, 45–18 (1959).

    Google Scholar 

  24. Sobolevskii, P. E., On the smoothness of generalized solutions of the Navier-Stokes equation, ibid Nauk USSR 131, 758–760 (1960).

Download references

Author information

Authors and Affiliations

Authors

Additional information

Communicated by R. Finn

Dedicated to Charles Loewner on the occasion of his 70th birthday

This work was supported in part by Office of Naval Research Contract Nonr-225(11) at Stanford University, while Fujita was on leave from Tokyo University. Reproduction in whole or in part is permitted for any purpose of the United States Government.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Fujita, H., Kato, T. On the Navier-Stokes initial value problem. I. Arch. Rational Mech. Anal. 16, 269–315 (1964). https://doi.org/10.1007/BF00276188

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00276188

Keywords

Navigation