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 \({\textbf {curl}}\).
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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.
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Communicated by E. Feireisl.
Dedicated to the memory of Antonín Novotný.
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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
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DOI: https://doi.org/10.1007/s00021-022-00728-w