Global Regularity Criterion for the 3D Navier-Stokes Equations Involving One Entry of the Velocity Gradient Tensor

Archive for Rational Mechanics and Analysis

Volumn 202, Issue 3, 2011, Pages 919-932

  Cao, Chongsheng ^a,b   Titi, Edriss S ^c  

^a FLORIDA INTERNATIONAL UNIVERSITY   (United States)

^b University of California Irvine   (United States)

^c WEIZMANN INSTITUTE OF SCIENCE   (Israel)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 81555209707     PISSN: 00039527     EISSN: 14320673     Source Type: Journal    
DOI: 10.1007/s00205-011-0439-6     Document Type: Article

References (39)

1. Adams R. A.: Sobolev Spaces. Academic Press, New York (1975).
2. Berselli L. C.: a regularity criterion for the solutions to the 3D Navier-Stokes equations. Differential Integral Equations 15, 1129-1137 (2002).
3. Berselli L. C., Galdi G. P.: Regularity criteria involving the pressure for the weak solutions to the Navier-Stokes equations. Proc. Am. Math. Soc 130, 3585-3595 (2002).
4. Cao C., Titi E. S.: Regularity criteria for the three-dimensional Navier-Stokes equations. Indiana Univ. Math. J. 57, 2643-2662 (2008).
5. Chae D., Lee J.: Regularity criterion in terms of pressure for the Navier-Stokes equations. Nonlinear Anal. 46, 727-735 (2001).
6. Constantin P.: A few results and open problems regarding incompressible fluids. Notices Amer. Math. Soc. 42, 658-663 (1995).
7. Constantin P., Foias C.: Navier-Stokes Equations. The University of Chicago Press, Chicago (1988).
8. Da Veiga H. B.: A sufficient condition on the pressure for the regularity of weak solutions to the Navier-Stokes equations. J. Math. Fluid Mech. 2, 99-106 (2000).
9. Doering C., Gibbon J.: Applied Analysis of the Navier-Stokes Equations. Cambridge University Press, Cambridge (1995).
10. 3, ∞-solutions of the Navier-Stokes equations and backward uniqueness. Russ. Math. Surv. 58, 211-250 (2003).
11. Fujita H., Kato T.: On the Navier-Stokes initial value problem. I. Arch. Rat. Mech. Anal. 3, 269-315 (1964).
12. Galdi G. P.: An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Vol. I, II. Springer, New York (1994).
13. p and regularity of weak solutions of the Navier-Stokes system. J. Differential Equations 62, 186-212 (1986).
14. r of the Navier-Stokes initial value problem. Arch. Rational Mech. Anal. 89, 267-281 (1985).
15. He C.: New sufficient conditions for regularity of solutions to the Navier-Stokes equations. Adv. Math. Sci. Appl. 12, 535-548 (2002).
16. m, with applications to weak solutions. Math. Z. 187, 471-480 (1984).
17. Kukavica I.: Role of the pressure for validity of the energy equality for solutions of the Navier-Stokes equation. J. Dyn. Diff. Equ. 18, 461-482 (2006).
18. Kukavica I., Ziane M.: One component regularity for the Navier-Stokes equation. Nonlinearity 19(2), 453-469 (2006).
19. Ladyzhenskaya, O. A.: Mathematical Theory of Viscous Incompressible Flow. Gordon and Breach, New York, English translation, 2nd ed., 1969.
20. Ladyzhenskaya O. A.: The Boundary Value Problems of Mathematical Physics. Springer, Berlin (1985).
21. Ladyzhenskaya, O. A.: The sixth millennium problem: Navier-Stokes equations, existence and smoothness. (Russian) Uspekhi Mat. Nauk. 58(2), 45-78 (2003); translation in Russian Math. Surveys58(2), 251-286 (2003).
22. Lemarié-Rieusset P. G.: Recent Developments in the Navier-Stokes Problem. Chapman & Hall, London (2002).
23. Leray J.: Sur le mouvement dun liquide visquex emplissant lespace. Acta Math. 63, 193-248 (1934).
24. Lions J. L.: Quelques Méthodes De Résolution Des Problèmes Aux Limites Non Linéaires. Dunod, Paris (1969).
25. Lions P. L.: Mathematical Topics in Fluid Mechanics: Volume 1: Incompressible Models. Oxford University Press, Oxford (1996).
26. Pokorný M.: On the result of He concerning the smoothness of solutions to the Navier-Stokes equations. Electron. J. Diff. Equ. 11, 1-8 (2003).
27. Pokorný M., Zhou Y.: On the regularity of the solutions of the Navier-Stokes equations via one velocity component. Nonlinearity 23, 1097-1107 (2010).
28. Prodi G.: Un teorema di unicità per le equazioni di Navier-Stokes. Ann. Mat. Pura Appl. 48, 173-182 (1959).
29. Raugel G., Sell G. R.: Navier-Stokes equations on thin 3D domains. I. Global attractors and global regularity of solutions. J. Am. Math. Soc. 6, 503-568 (1993).
30. Seregin G., Sverák V.: Navier-Stokes equations with lower bounds on the pressure. Arch. Ration. Mech. Anal. 163(1), 65-86 (2002).
31. Serrin J.: On the interior regularity of weak solutions of the Navier-Stokes equations. Arch. Rational Mech. Anal. 9, 187-191 (1962).
32. Sohr H.: The Navier-Stokes Equations, An Elementary Functional Analytic Approach. Birkhäuser Verlag, Basel (2001).
33. Sohr H.: A regularity class for the Navier-Stokes equations in Lorentz spaces. J. Evol. Equ. 1, 441-467 (2001).
34. Sohr R.: A generalization of Serrin's regularity criterion for the Navier-Stokes equations. Quad. Mat. 10, 321-347 (2002).
35. Temam, R.: Navier-Stokes Equations, Theory and Numerical Analysis. North-Holland, 1984.
36. Temam, R.: Navier-Stokes Equations and Nonlinear Functional Analysis. 2nd edn, SIAM, 1995.
37. Temam, R.: Some developments on Navier-Stokes equations in the second half of the 20th century. Development of Mathematics 1950-2000. Birkhauser, Basel, 1049-1106, 2000.
38. Zhou Y.: A new regularity criterion for the Navier-Stokes equations in terms of the gradient of one velocity component. Methods Appl. Anal. 9, 563-578 (2002).
39. 3. Proc. Amer. Math. Soc. 134, 149-156 (2005).