Remarks on the Global Existence for Incompressible Navier-Stokes Equations

Abstract

In this article, the authors use the special structure of helicity for the three-dimensional incompressible Navier-Stokes equations to construct a family of finite energy smooth solutions to the Navier-Stokes equations which critical norms can be arbitrarily large.

Keywords

Navier-Stokes euqations / Helicity / Global existence / Critical norm

References

[1] Bourgain J, Pavlovì N. Ill-posedness of the Navier-Stokes equations in a critical space in 3D. J. Funct. Anal., 2008, 255: 2233-2247

[2] Cannone M, Meyer Y, Planchon F. Solutions auto-similarities to the des équations de Navier-Stokes. Séminaire sur les Équations aux Dérivées Partielles, 1993 1994

[3] Fefferman, C. L., http://www.claymath.org/millennium/Navier-Stokes-Equations/, Accessed 1 May, 2000.

[4] Fujita H, Kato T. On the Navier-Stokes initial value problem I. Arch. Rational Mech. Anal., 1964, 16: 269-315

[5] Giga Y, Miyakawa T. Solutions in L ^r of the Navier-Stokes initial value problem. Arch. Rational Mech. Anal.., 1985, 89: 267-281

[6] Kato T. Strong L ^p-solutions of the Navier-Stokes equation in ℝⁿ. Math. Z., 1984, 187: 471-480

[7] Koch H, Tataru D. Well-posedness for the Navier-Stokes equations. Adv. Math, 2001, 157: 22-35

[8] Lei Z, Lin F H, Zhou Y. Structure of helicity and global solutions of incompressible Navier-Stokes equation. Arch. Rational. Mech. Anal., 2015, 218: 1417-1430

[9] Lin, Y. R., Zhang, H. L. and Zhou, Y., Global smooth solutions of MHD equations with large data, arXiv:1512, 00646.

[10] Robinson J C, Rodrigo J L, Sadowski W. The Three-Dimensional Navier-Stokes Equations, 2016, Cambridge: Cambridge University Press