Archive for Rational Mechanics and Analysis

, Volume 104, Issue 3, pp 223–250

Two-dimensional Navier-Stokes flow with measures as initial vorticity

  • Yoshikazu Giga
  • Tetsuro Miyakawa
  • Hirofumi Osada
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References

  1. 1.
    Aronson, D. G., Bounds for the fundamental solution of a parabolic equation. Bull. Amer. Math. Soc. 73, 890–896 (1968).Google Scholar
  2. 2.
    Aronson, D. G., & J. Serrin, Local behavior of solutions of quasilinear parabolic equations. Arch. Rational Mech. Anal. 25, 81–122 (1967).Google Scholar
  3. 3.
    Benfatto, G., Esposito, R., & M. Pulvirenti, Planar Navier-Stokes flow for singular initial data. Nonlinear Anal. 9, 533–545 (1985).Google Scholar
  4. 4.
    Bergh, J., & J. Löfström, Interpolation Spaces, An Introduction. Berlin Heidelberg New York: Springer-Verlag 1976.Google Scholar
  5. 5.
    Brézis, H., & A. Friedman, Nonlinear parabolic equations involving measures as initial data. J. Math. Pures et appl. 62, 73–97 (1983).Google Scholar
  6. 6.
    Dobrushin, R. L., Prescribing a system of random variables by conditional distributions. Theory Prob. Appl. 15, 458–486 (1970).Google Scholar
  7. 7.
    Fabes, E. B., Jones, B. F., & N. M. Rivière, The initial value problem for the Navier-Stokes equations with data in L p Arch. Rational Mech. Anal. 45, 222–240 (1972).Google Scholar
  8. 8.
    Friedman, A., Partial Differential Equations of Parabolic Type. New Jersey: Prentice-Hall 1964.Google Scholar
  9. 9.
    Friedman, A., Partial Differential Equations. New York: Holt, Rinehart & Winston 1969.Google Scholar
  10. 10.
    Fujita, H., & T. Kato, On the Navier-Stokes initial value problem I. Arch. Rational Mech. Anal. 16, 269–315 (1964).Google Scholar
  11. 11.
    Giga, Y., & T. Miyakawa, Solutions in L r of the Navier-Stokes initial value problem. Arch. Rational Mech. Anal. 89, 267–281 (1985).Google Scholar
  12. 12.
    Giga, Y., Solutions for semilinear parabolic equations in L p and regularity of weak solutions of the Navier-Stokes system. J. Differential Equations 62, 186–212 (1986).Google Scholar
  13. 13.
    Gilbarg, D., & N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed. Berlin Heidelberg New York: Springer-Verlag 1983.Google Scholar
  14. 14.
    Kato, T., Strong Lp-solutions of the Navier-Stokes equation in R m, with applications to weak solutions. Math. Z. 187, 471–480 (1984).Google Scholar
  15. 15.
    Kato, T., Remarks on the Euler and Navier-Stokes equations in R 2. Nonlinear Functional Analysis and its Applications, F. E. Browder ed., Proc. of Symposia in Pure Math. 45, part 2, 1–8. Providence, RI: Amer. Math. Soc. 1986.Google Scholar
  16. 16.
    Ladyzhenskaya, O. A., The Mathematical Theory of Viscous Incompressible Flow. New York: Gordon & Breach 1969.Google Scholar
  17. 17.
    Leray, J., Etude de diverses équations intégrales non linéaires et de quelques problèmes que pose l'hydrodynamique. J. Math. pures et appl., Serie 9, 12, 1–82 (1933).Google Scholar
  18. 18.
    Liu, T.-P., & M. Pierre, Source-solutions and asymptotic behavior in conservation laws. J. Differential Equations 51, 419–441 (1984).Google Scholar
  19. 19.
    Marchioro, C., & M. Pulvirenti, Hydrodynamics in two dimensions and vortex theory. Commun. Math. Phys. 84, 483–503 (1982).Google Scholar
  20. 20.
    Marchioro, C., & M. Pulvirenti, Euler evolution for singular initial data and vortex theory. Commun. Math. Phys. 91, 563–572 (1983).Google Scholar
  21. 21.
    McGrath, F. J., Nonstationary planar flow of viscous and ideal fluids. Arch. Rational Mech. Anal. 27, 329–348 (1968).Google Scholar
  22. 22.
    McKean, H. P., Jr., Propagation of chaos for a class of nonlinear parabolic equations. Lecture series in differential equations, Session 7: Catholic Univ. 1967.Google Scholar
  23. 23.
    Niwa, Y., Semilinear heat equations with measures as initial data. Preprint.Google Scholar
  24. 24.
    Osada, H., & S. Kotani, Propagation of chaos for the Burgers equation. J. Math. Soc. Japan 37, 275–294 (1985).Google Scholar
  25. 25.
    Osada, H., Diffusion processes with generators of generalized divergence form. J. Math. Kyoto Univ. 27, 597–619 (1987).Google Scholar
  26. 26.
    Osada, H., Propagation of chaos for the two dimensional Navier-Stokes equations. Probabilistic Methods in Math. Phys., K. Ito & N. Ikeda eds., 303–334, Tokyo: Kinokuniya 1987.Google Scholar
  27. 27.
    Ponce, G., On two dimensional incompressible fluids. Commun. Partial Differ. Equations 11, 483–511 (1986).Google Scholar
  28. 28.
    Reed, M., & B. Simon, Methods of Modern Mathematical Physics Vol. I, II; New York: Academic Press 1972, 1975.Google Scholar
  29. 29.
    Sznitman, A. S., Propagation of chaos result for the Burgers equation. Probab. Th. Rel. Fields 71, 581–613 (1986).Google Scholar
  30. 30.
    Temam, R., Navier-Stokes Equations. Amsterdam: North-Holland 1977.Google Scholar
  31. 31.
    Turkington, B., On the evolution of a concentrated vortex in an ideal fiuid. Arch. Rational Mech. Anal. 97, 75–87 (1987).Google Scholar
  32. 32.
    Wahl, W. von, The Equations of Navier-Stokes and Abstract Parabolic Equations. Braunschweig: Vieweg Verlag 1985.Google Scholar
  33. 33.
    Weissler, F. B., The Navier-Stokes initial value problem in L p. Arch. Rational Mech. Anal. 74, 219–230 (1980).Google Scholar
  34. 34.
    Kato, T., & G. Ponce, Well-posedness of the Euler and Navier-Stokes equations in the Lebesgue spaces L sp(R 2). Rev. Mat. Iberoamericana 2, 73–88 (1986).Google Scholar
  35. 35.
    Baras, P., & M. Pierre, Problèmes paraboliques semi-linéaires avec données mesures. Applicable Analysis 18, 111–149 (1984).Google Scholar
  36. 36.
    DiPerna, R. J., & A. J. Majda, Concentrations in regularizations for 2-D incompressible flow. Comm. Pure Appl. Math. 16, 301–345 (1987).Google Scholar
  37. 37.
    Cottet, G.-H., Équations de Navier-Stokes dans le plan avec tourbillon initial mesure. C. R. Acad. Sc. Ser. 1, 303, 105–108 (1986).Google Scholar

Copyright information

© Springer-Verlag 1988

Authors and Affiliations

  • Yoshikazu Giga
    • 1
    • 2
    • 3
  • Tetsuro Miyakawa
    • 1
    • 2
    • 3
  • Hirofumi Osada
    • 1
    • 2
    • 3
  1. 1.Department of MathematicsHokkaido UniversitySapporo
  2. 2.Department of MathematicsHiroshima UniversityHiroshima
  3. 3.Department of MathematicsNara Women's UniversityNara

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