Archive for Rational Mechanics and Analysis

, Volume 202, Issue 3, pp 919–932

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

Article

Abstract

In this paper we provide a sufficient condition, in terms of only one of the nine entries of the gradient tensor, that is, the Jacobian matrix of the velocity vector field, for the global regularity of strong solutions to the three-dimensional Navier–Stokes equations in the whole space, as well as for the case of periodic boundary conditions.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Adams R.A.: Sobolev Spaces. Academic Press, New York (1975)MATHGoogle Scholar
  2. Berselli L.C.: a regularity criterion for the solutions to the 3D Navier–Stokes equations. Differential Integral Equations 15, 1129–1137 (2002)MathSciNetMATHGoogle Scholar
  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)MathSciNetMATHCrossRefGoogle Scholar
  4. Cao C., Titi E.S.: Regularity criteria for the three-dimensional Navier–Stokes equations. Indiana Univ. Math. J. 57, 2643–2662 (2008)MathSciNetMATHCrossRefGoogle Scholar
  5. Chae D., Lee J.: Regularity criterion in terms of pressure for the Navier-Stokes equations. Nonlinear Anal. 46, 727–735 (2001)MathSciNetMATHCrossRefGoogle Scholar
  6. Constantin P.: A few results and open problems regarding incompressible fluids. Notices Amer. Math. Soc. 42, 658–663 (1995)MathSciNetMATHGoogle Scholar
  7. Constantin P., Foias C.: Navier–Stokes Equations. The University of Chicago Press, Chicago (1988)MATHGoogle Scholar
  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)MathSciNetMATHCrossRefGoogle Scholar
  9. Doering C., Gibbon J.: Applied Analysis of the Navier–Stokes Equations. Cambridge University Press, Cambridge (1995)MATHCrossRefGoogle Scholar
  10. Escauriaza L., Seregin G.A., Sverak V.: L 3, ∞-solutions of the Navier–Stokes equations and backward uniqueness. Russ. Math. Surv. 58, 211–250 (2003)MathSciNetCrossRefGoogle Scholar
  11. Fujita H., Kato T.: On the Navier-Stokes initial value problem. I. Arch. Rat. Mech. Anal. 3, 269–315 (1964)MathSciNetCrossRefGoogle Scholar
  12. Galdi G.P.: An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Vol. I, II. Springer, New York (1994)CrossRefGoogle Scholar
  13. 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)MathSciNetCrossRefGoogle Scholar
  14. Giga Y., Miyakawa T.: Solutions in L r of the Navier–Stokes initial value problem. Arch. Rational Mech. Anal. 89, 267–281 (1985)MathSciNetADSMATHCrossRefGoogle Scholar
  15. He C.: New sufficient conditions for regularity of solutions to the Navier–Stokes equations. Adv. Math. Sci. Appl. 12, 535–548 (2002)MathSciNetMATHGoogle Scholar
  16. Kato T.: Strong L p solutions of the Navier–Stokes equation in R m, with applications to weak solutions. Math. Z. 187, 471–480 (1984)MathSciNetMATHCrossRefGoogle Scholar
  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)MathSciNetMATHCrossRefGoogle Scholar
  18. Kukavica I., Ziane M.: One component regularity for the Navier–Stokes equation. Nonlinearity 19(2), 453–469 (2006)MathSciNetADSMATHCrossRefGoogle Scholar
  19. Ladyzhenskaya, O.A.: Mathematical Theory of Viscous Incompressible Flow. Gordon and Breach, New York, English translation, 2nd ed., 1969Google Scholar
  20. Ladyzhenskaya O.A.: The Boundary Value Problems of Mathematical Physics. Springer, Berlin (1985)MATHGoogle Scholar
  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. Surveys 58(2), 251–286 (2003)Google Scholar
  22. Lemarié–Rieusset P.G.: Recent Developments in the Navier–Stokes Problem. Chapman & Hall, London (2002)MATHCrossRefGoogle Scholar
  23. Leray J.: Sur le mouvement dun liquide visquex emplissant lespace. Acta Math. 63, 193–248 (1934)MathSciNetMATHCrossRefGoogle Scholar
  24. Lions J.L.: Quelques Méthodes De Résolution Des Problèmes Aux Limites Non Linéaires. Dunod, Paris (1969)MATHGoogle Scholar
  25. Lions P.L.: Mathematical Topics in Fluid Mechanics: Volume 1: Incompressible Models. Oxford University Press, Oxford (1996)Google Scholar
  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)Google Scholar
  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)MathSciNetADSMATHCrossRefGoogle Scholar
  28. Prodi G.: Un teorema di unicità per le equazioni di Navier–Stokes. Ann. Mat. Pura Appl. 48, 173–182 (1959)MathSciNetMATHCrossRefGoogle Scholar
  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)MathSciNetMATHGoogle Scholar
  30. Seregin G., Sverák V.: Navier–Stokes equations with lower bounds on the pressure. Arch. Ration. Mech. Anal. 163(1), 65–86 (2002)MathSciNetMATHCrossRefGoogle Scholar
  31. Serrin J.: On the interior regularity of weak solutions of the Navier–Stokes equations. Arch. Rational Mech. Anal. 9, 187–191 (1962)MathSciNetADSMATHCrossRefGoogle Scholar
  32. Sohr H.: The Navier–Stokes Equations, An Elementary Functional Analytic Approach. Birkhäuser Verlag, Basel (2001)MATHGoogle Scholar
  33. Sohr H.: A regularity class for the Navier–Stokes equations in Lorentz spaces. J. Evol. Equ. 1, 441–467 (2001)MathSciNetMATHCrossRefGoogle Scholar
  34. Sohr R.: A generalization of Serrin’s regularity criterion for the Navier–Stokes equations. Quad. Mat. 10, 321–347 (2002)MathSciNetGoogle Scholar
  35. Temam, R.: Navier–Stokes Equations, Theory and Numerical Analysis. North–Holland, 1984Google Scholar
  36. Temam, R.: Navier–Stokes Equations and Nonlinear Functional Analysis. 2nd edn, SIAM, 1995Google Scholar
  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, 2000Google Scholar
  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)MathSciNetMATHGoogle Scholar
  39. Zhou Y.: On regularity criteria in terms of pressure for the Navier–Stokes equations in R 3. Proc. Amer. Math. Soc. 134, 149–156 (2005)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Department of MathematicsFlorida International UniversityMiamiUSA
  2. 2.Department of Mathematics and Department of Mechanical and Aerospace EngineeringUniversity of CaliforniaIrvineUSA
  3. 3.Department of Computer Science and Applied MathematicsWeizmann Institute of ScienceRehovotIsrael

Personalised recommendations