On Leray's self-similar solutions of the Navier-Stokes equations satisfying local energy estimates

Archive for Rational Mechanics and Analysis

Volumn 143, Issue 1, 1998, Pages 29-51

  Tsai, Tai Peng ^a  

^a UNIVERSITY OF MINNESOTA   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0032363760     PISSN: 00039527     EISSN: None     Source Type: Journal    
DOI: 10.1007/s002050050099     Document Type: Article

References (44)

1. S. ANGENENT, T. ILMANEN & J. J. L. VELÁZQUEZ, preprint.
2. L. CAFFARELLI, R.V. KOHN & L. NIRENBERG, Partial regularity of suitable weak solution of the Navier-Stokes equations, Comm. Pure Appl. Math. 35 (1982), 771-831.
3. A. P. CALDERÓN & A. ZYGMUND, On existenct of certain singular integrals, Acta Math. 88 (1952), 85-139.
4. 3, Comm. Part. Diff. Eqs. 21 (1996), 179-193.
5. L. CATTABRIGA, Su un problema al retorno relativo al sistema di equazioni di Stokes, Rend. Sem. Math. Univ. Padova 31 (1961), 308-340.
6. P. CONSTANTIN & C. FEFFERMAN, Direction of vorticity and the problem of global regularity for the Navier-Stokes equations, Indiana Univ. Math. J. 42 (1993), 775-789.
7. P. CONSTANTIN & C. FOIAS, Navier-Stokes equations, Univ. of Chicago Press, 1988.
8. p, Arch. Rational Mech. Anal. 45 (1972), 222-240.
9. p spaces of several variables, Acta Math. 129 (1972), 137-193.
10. J. FREHSE & M. RŮŽIČKA, On the regularity of the stationary Navier-Stokes equations, Ann. Sc. Norm. Pisa 21 (1994), 63-95.
11. J. FREHSE & M. RŮŽIČKA, Regularity of the stationary Navier-Stokes equations in bounded domains. Arch. Rational Mech. Anal. 128 (1994), 361-381.
12. G. GALDI, An introduction to the mathematical theory of Navier-Stokes equations, I, II, Springer, 1994.
13. M. GIAQUINTA & G. MODICA, Non linear systems of the type of the stationary Navier-Stokes equations, J. Reine Angew. Math. 330 (1982), 173-214.
14. p and regularity of weak solutions of the Navier-Stokes system, J. Diff. Equations 61 (1986), 186-212.
15. Y. GIGA, On elliptic equations related to self-similar solutions for nonlinear heat equations, Hiroshima Math. J. 16 (1986), 539-552.
16. Y. GIGA & R. V. KOHN, Asymptotically self-similar blow-up of semilinear heat equations, Comm. Pure Appl. Math. 38 (1985), 297-319.
17. 3 with measures as initial vorticity and Morrey spaces, Comm. in Part. Diff. Eqs. 14 (1989), 577-618.
18. D. GILBARG & H. F. WEINBERGER, Asymptotic properties of Leray's solution of the stationary two-dimensional Navier-Stokes equations, Russian Math. Surveys, 29 (1974), 109-123.
19. +, Pacific J. Math. 164 (1994), 351-359.
20. E. HOPF, Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen, Math. Nachr. 4 (1951), 213-231.
21. F. JOHN & L. NIRENBERG, On functions of bounded mean oscillation, Comm. Pure Appl. Math. 14 (1961), 415-426.
22. N. KOPELL & M. LANDMAX, Spatial structure of the focusing singularity of the nonlinear Schrödinger equation: a geometrical analysis, SIAM. J. Appl. Math. 55 (1995), 1297-1323.
23. O. A. LADYZHENSKAYA, The mathematical theory of viscous incompressible flow, 2nd ed., Gordon and Breach, 1969.
24. J. LERAY, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math. 63 (1934), 193-248.
25. F. H. LIN & C. LIU, Partial regularity of the dynamic system modeling the flow of liquid crystals, Discrete Contin. Dynam. System 2 (1996), 1-22.
26. J. NEČAS, M. RŮŽIČKA & V. ŠVERÁK, On Leray's self-similar solutions of the Navier-Stokes equations, Acta Math. 176 (1996), 283-294.
27. J. MÁLEK, J. NEČAS, M. POKORNÝ & M. E. SCHONBEK, On possible singular solutions to the Navier-Stokes equations, preprint.
28. T. OHYAMA, Interior regularity of weak solutions to the Navier-Stokes equations, Proc. Japan Acad. 36 (1960), 273-277.
29. V. SCHEFFER, Hausdorff measure and the Navier-Stokes equations, Comm. Math. Phys. 55 (1977), 97-112.
30. V. SCHEFFER, A solution to the Navier-Stokes inequality with an internal singularity, Comm. Math. Phys. 101 (1985), 47-85.
31. J. SERRIN, On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal. 9 (1962), 187-195.
32. J. SERRIN, Mathematical principles of classical fluid mechanics, in Handbuch der Physik, vol. 8, part 1, pp. 125-263, Springer, Berlin, Göttingen, Heidelberg, 1959.
33. H. SOHR & W. VON WAHL, On the regularity of the pressure of weak solutions of Navier-Stokes Equations, Arch. Math. 46 (1986), 428-439.
34. E. M. STEIN, Singular integrals and the differentiability properties of functions, Princeton University Press, 1970.
35. E. M. STEIN, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton University Press, 1993.
36. M. STRUWE, On the evolution of harmonic maps in higher dimensions, J. Diff. Geom. 28 (1988), 485-502.
37. M. STRUWE, On partial regularity results for the Navier-Stokes equations, Comm. Pure Appl. Math. 41 (1988), 437-458.
38. 5, Math. Ann. 302 (1995), 719-741.
39. V. ŠVERÁK & T.-P. TSAI, to appear.
40. S. TAKAHASHI, On interior regularity criteria for weak solutions of the Navier-Stokes equations, Manuscripta Math. 69 (1990), 237-254.
41. R. TEMAM, Navier-Stokes equations: theory and numerical analysis, North-Holland, Amsterdam and New York, 1977.
42. G. TIAN & Z. XIN, Gradient estimation on Navier-Stokes equations, preprint.
43. A. VALLI, On the integral representation of the solution to the Stokes svstem, Rend. Sem. Mat. Univ. Padova 74 (1985), 85-114.
44. W. VON WAHL, The equations of Navier-Stokes and abstract parabolic equations, Vieweg, 1985.