Abstract
We give some regularity criterion (of a weak\(-L^{p}\) Serrin type) of a weak solution to the 3D Navier–Stokes equations in a bounded domain \(\Omega \subset \mathbb {R}^3\) with a smooth boundary. In particular, in case of the half space, we give a regularity condition of a weak solution with respect to a tangential component of the velocity flow vector.
Similar content being viewed by others
References
Bae, H.-O., Choe, H.J.: A regularity criterion for the Navier–Stokes equations. Comm. Partial Differ. Equ. 32, 1173–1187 (2007)
Bae, H.-O., Wolf, J.: A local regularity condition involving two velocity components of Serrin-type for the Navier–Stokes equations. C. R. Math. Acad. Sci. Paris 354, 167–174 (2016)
Beirão da Veiga, H., Berselli, L.C.: Navier–Stokes equations: Green’s matrices, vorticity direction, and regularity up to the boundary. J. Differ. Equ. 246, 597–628 (2009)
Beirão da Veiga, H., Crispo, F.: Sharp inviscid limit results under Navier type boundary conditions. An \(L^p\) theory. J. Math. Fluid Mech. 12, 397–411 (2010)
Berselli, L.C., Spirito, S.: On the vanishing viscosity limit of 3D Navier–Stokes equations under slip boundary conditions in general domains. Comm. Math. Phys. 316, 171–198 (2012)
Bosia, S., Pata, V., Robinson, J.C.: A weak-\(L^p\) Prodi–Serrin type regularity criterion for the Navier–Stokes equations. J. Math. Fluid Mech. 16, 721–725 (2014)
Chae, D.: On the regularity conditions of suitable weak solutions of the 3D Navier–Stokes equations. J. Math. Fluid Mech. 12, 171–180 (2010)
Chae, D., Choe, H.J.: Regularity of solutions to the Navier–Stokes equation. Electron. J. Differ. Equ. 5, 1–7 (1999)
Guo, Z., Kučera, P., Skalák, Z.: Regularity criterion for solutions to the Navier–Stokes equations in the whole 3D space based on two vorticity components. J. Math. Anal. Appl. 458, 755–766 (2018)
Hopf, E.: Über die Anfangswertaufgabe für die hydrodynamischen Grundglei-chungen. Math. Nachr. 4, 213–231 (1951)
Ji, X., Wang, Y., Wei, W.: New regularity criteria based on pressure or gradient of velocity in Lorentz spaces for the 3D Navier–Stokes equations. J. Math. Fluid Mech. 22(1), Paper No. 13, 8 pp. (2020)
Kang, K., Kim, J.-M.: Regularity criteria of the magenetohydrodynamic equations in bounded domains or a half space. J. Differ. Equ. 253, 764–794 (2012)
Kukavica, I., Ziane, M.: Navier–Stokes equations with regularity in one direction. J. Math. Phys. 48, 065203, 10 pp (2007)
Ladyženskaja, O.A., Solonnikov, V.A., Uralceva, N.N.: Linear and Quasilinear Equations of Parabolic Type. Translated from the Russian by S. Smith. Translations of Mathematical Monographs, vol. 23, American Mathematical Society, Providence, RI (1967)
Lee, J.: Notes on the geometric regularity criterion of 3D Navier–Stokes system. J. Math. Phys. 53, (2012)
Leray, J.: Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math. 63, 193–248 (1934)
O’Neil, R.: Convolution operators and \(L(p, q)\) spaces. Duke Math. J. 30, 129–142 (1963)
Prodi, G.: Un teorema di unicita per le equazioni di Navier–Stokes. Ann. Mat. Pura Appl. 48, 173–182 (1959)
Serrin, J.: On the interior regularity of weak solutions of the Navier–Stokes equations. Arch. Rational Mech. Anal. 9, 187–195 (1962)
Skalák, Z.: Commentary on local and boundary regularity of weak solutions to Navier–Stokes equations. Electron. J. Differ. Equ., No. 9, 14 pp. (2004)
Suzuki, T.: A remark on the regularity of weak solutions to the Navier–Stokes equations in terms of the pressure in Lorentz spaces. Nonlinear Anal. 75, 3849–3853 (2012)
Triebel, H.: Theory of Function Spaces. Birkhäuser Verlag, Basel (1983)
von Wahl, W.: Estimating \(\nabla u\) by \(\text{div} u\) and \({\rm {curl}}\,u\). Math. Methods Appl. Sci. 15, 123–143 (1992)
Wang, W., Zhang, L., Zhang, Z.: On the interior regularity criteria of the 3-D Navier–Stokes equations involving two velocity components. Discrete Contin. Dyn. Syst. 38, 2609–2627 (2018)
Acknowledgements
The author thanks the knowledgeable referee for his/her valuable comments and helpful suggestions. Jae-Myoung Kim was supported by the National Research Foundation of Korea Grant funded by the Korean Government (NRF-2020R1C1C1A01006521).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Kim, JM. Some regularity criteria of a weak solution to the 3D Navier–Stokes equations in a domain. Arch. Math. 117, 215–225 (2021). https://doi.org/10.1007/s00013-021-01613-0
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00013-021-01613-0