Abstract
We deal with a weak solution \({{\textbf {v}}}\) to the Navier–Stokes initial value problem in \({{\mathbb {R}}}^3\times (0,T)\), that satisfies the strong energy inequality. We impose conditions on certain spectral projections of \({\varvec{\omega }}:={\textbf {curl}}\, {{\textbf {v}}}\) or just \({{\textbf {v}}}\), and we prove the regularity of solution \({{\textbf {v}}}\). The spectral projection is defined by means of the spectral resolution of identity associated with the self–adjoint operator \({\textbf {curl}}\).
Similar content being viewed by others
References
Beirão da Veiga, H.: Concerning the regularity problem for the solutions of the Navier–Stokes equations. C.R. Acad. Sci. Paris 321(Série I), 405–408 (1995)
Birman, M.S., Solomyak, M.Z.: Spectral Theory of Self-Adjoint Operators in Hilbert Space. D. Reidel Publishing Company, Dordrecht (1987)
Caffarelli, L., Kohn, R., Nirenberg, L.: Partial regularity of suitable weak solutions of the Navier–Stokes equations. Commun. Pure Appl. Math. 35, 771–831 (1982)
Chae, D., Choe, H. J.: Regularity of solutions to the Navier–Stokes equations. Electronic J. Differ. Equ., 7 (1999)
Farwig, R., Neustupa, J.: On the spectrum of a Stokes-type operator arising from flow around a rotating body. Manuscripta Math. 122, 419–437 (2007)
Galdi, G.P.: An Introduction to the Mathematical Theory of the Navier–Stokes Equations, 2nd edn. Springer, Berlin (2011)
Galdi, G.P.: An Introduction to the Navier–Stokes initial–boundary value problem. In: Galdi, G.P., Heywood, J., Rannacher, R. (eds.) In: Fundamental Directions in Mathematical Fluid Mechanics. “Advances in Mathematical Fluid Mechanics”, pp. 1–98. Birkhauser, Basel (2000)
Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin (1966)
Ladyzhenskaya, O.A., Seregin, G.A.: On partial regularity of suitable weak solutions to the three-dimensional Navier–Stokes equations. J. Math. Fluid Mech. 1, 356–387 (1999)
Leray, J.: Sur le mouvements d’un liquide visqueux emplissant l’espace. Acta Math. 63, 193–248 (1934)
Neustupa, J., Penel, P.: Regularity of a weak solution to the Navier–Stokes equations via one component of a spectral projection of vorticity. SIAM J. Math. Anal. 46(2), 1681–1700 (2014)
Picard, R.: On a selfadjoint realization of \({{\bf curl}}\) in exterior domain. Math. Zeitschrift 229, 319–338 (1998)
Runst, T., Sickel, W.: Sobolev Spaces of Fractionalk Order, Nemytskij Operators, and Nonlinear Partial Differential Equations. Walter de Gruyter, Berlin (1996)
Serrin, J.: On the interior regularity of weak solutions of the Navier–Stokes equations. Arch. Rat. Mech. Anal. 9, 187–195 (1962)
Seregin, G.A.: A certain necessary condition of potential blow up for Navier–Stokes equations. Commun. Math. Phys. 312, 833–845 (2012)
Sohr, H.: The Navier-Stokes Equations. An Elementary Functional Analytic Approach., Birkhäuser Advanced Texts, Berlin (2001)
Yosida, Z., Giga, Y.: Remarks on spectra of operator \({{\bf rot}}\). Math. Zeitschrift 204, 235–245 (1990)
Acknowledgements
The first author has been supported by the Grant Agency of the Czech Republic, grant No. 22-01591S, and the Academy of Sciences of the Czech Republic (RVO 67985840). The third author has been supported by the National Research Foundation of Korea (NRF), grant No. 2021R1A2C4002840.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
On behalf of all authors, the corresponding author states that there is no conflict of interest.
Additional information
Communicated by E. Feireisl.
Dedicated to the memory of Antonín Novotný.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This article is part of the topical collection “In memory of Antonin Novotny” edited by Eduard Feireisl, Paolo Galdi, and Milan Pokorny.
Rights and permissions
Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Neustupa, J., Penel, P. & Yang, M. Regularity Criteria for Weak Solutions to the Navier–Stokes Equations in Terms of Spectral Projections of Vorticity and Velocity. J. Math. Fluid Mech. 24, 104 (2022). https://doi.org/10.1007/s00021-022-00728-w
Accepted:
Published:
DOI: https://doi.org/10.1007/s00021-022-00728-w