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.
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References
Adams R.A.: Sobolev Spaces. Academic Press, New York (1975)
Berselli L.C.: a regularity criterion for the solutions to the 3D Navier–Stokes equations. Differential Integral Equations 15, 1129–1137 (2002)
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)
Cao C., Titi E.S.: Regularity criteria for the three-dimensional Navier–Stokes equations. Indiana Univ. Math. J. 57, 2643–2662 (2008)
Chae D., Lee J.: Regularity criterion in terms of pressure for the Navier-Stokes equations. Nonlinear Anal. 46, 727–735 (2001)
Constantin P.: A few results and open problems regarding incompressible fluids. Notices Amer. Math. Soc. 42, 658–663 (1995)
Constantin P., Foias C.: Navier–Stokes Equations. The University of Chicago Press, Chicago (1988)
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)
Doering C., Gibbon J.: Applied Analysis of the Navier–Stokes Equations. Cambridge University Press, Cambridge (1995)
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)
Fujita H., Kato T.: On the Navier-Stokes initial value problem. I. Arch. Rat. Mech. Anal. 3, 269–315 (1964)
Galdi G.P.: An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Vol. I, II. Springer, New York (1994)
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)
Giga Y., Miyakawa T.: Solutions in L r of the Navier–Stokes initial value problem. Arch. Rational Mech. Anal. 89, 267–281 (1985)
He C.: New sufficient conditions for regularity of solutions to the Navier–Stokes equations. Adv. Math. Sci. Appl. 12, 535–548 (2002)
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)
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)
Kukavica I., Ziane M.: One component regularity for the Navier–Stokes equation. Nonlinearity 19(2), 453–469 (2006)
Ladyzhenskaya, O.A.: Mathematical Theory of Viscous Incompressible Flow. Gordon and Breach, New York, English translation, 2nd ed., 1969
Ladyzhenskaya O.A.: The Boundary Value Problems of Mathematical Physics. Springer, Berlin (1985)
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)
Lemarié–Rieusset P.G.: Recent Developments in the Navier–Stokes Problem. Chapman & Hall, London (2002)
Leray J.: Sur le mouvement dun liquide visquex emplissant lespace. Acta Math. 63, 193–248 (1934)
Lions J.L.: Quelques Méthodes De Résolution Des Problèmes Aux Limites Non Linéaires. Dunod, Paris (1969)
Lions P.L.: Mathematical Topics in Fluid Mechanics: Volume 1: Incompressible Models. Oxford University Press, Oxford (1996)
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)
Pokorný M., Zhou Y.: On the regularity of the solutions of the Navier–Stokes equations via one velocity component. Nonlinearity 23, 1097–1107 (2010)
Prodi G.: Un teorema di unicità per le equazioni di Navier–Stokes. Ann. Mat. Pura Appl. 48, 173–182 (1959)
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)
Seregin G., Sverák V.: Navier–Stokes equations with lower bounds on the pressure. Arch. Ration. Mech. Anal. 163(1), 65–86 (2002)
Serrin J.: On the interior regularity of weak solutions of the Navier–Stokes equations. Arch. Rational Mech. Anal. 9, 187–191 (1962)
Sohr H.: The Navier–Stokes Equations, An Elementary Functional Analytic Approach. Birkhäuser Verlag, Basel (2001)
Sohr H.: A regularity class for the Navier–Stokes equations in Lorentz spaces. J. Evol. Equ. 1, 441–467 (2001)
Sohr R.: A generalization of Serrin’s regularity criterion for the Navier–Stokes equations. Quad. Mat. 10, 321–347 (2002)
Temam, R.: Navier–Stokes Equations, Theory and Numerical Analysis. North–Holland, 1984
Temam, R.: Navier–Stokes Equations and Nonlinear Functional Analysis. 2nd edn, SIAM, 1995
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
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)
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)
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Cao, C., Titi, E.S. Global Regularity Criterion for the 3D Navier–Stokes Equations Involving One Entry of the Velocity Gradient Tensor. Arch Rational Mech Anal 202, 919–932 (2011). https://doi.org/10.1007/s00205-011-0439-6
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DOI: https://doi.org/10.1007/s00205-011-0439-6