Abstract
We study the local regularity of solutions to the Navier–Stokes equations. We show for a suitable weak solution (u, p) on an open space-time domain D that if \( {\partial}_3u\in {L}_t^p{L}_x^q(D) \), where 2/p + 3/q = 2 and q ∈ (27/16, 5/2), then the solution is regular in D.
Similar content being viewed by others
References
L. Caffarelli, R. Kohn, and L. Nirenberg, Partial regularity of suitable weak solutions of the Navier–Stokes equations, Commun. Pure Appl. Math., 35(6):771–831, 1982.
C. Cao, Sufficient conditions for the regularity to the 3D Navier–Stokes equations, Discrete Contin. Dyn. Syst., 26(4): 1141–1151, 2010.
C. Cao and E.S. Titi, Global regularity criterion for the 3D Navier–Stokes equations involving one entry of the velocity gradient tensor, Arch. Ration. Mech. Anal., 202(3):919–932, 2011.
J.-Y. Chemin, I. Gallagher, and P. Zhang, Some remarks about the possible blow-up for the Navier–Stokes equations, Commun. Partial Differ. Equations, 44(12):1387–1405, 2019.
J.-Y. Chemin, P. Zhang, and Z. Zhang, On the critical one component regularity for 3-D Navier–Stokes system: General case, Arch. Ration. Mech. Anal., 224(3):871–905, 2017.
L. Escauriaza, G. Seregin, and V. Šverák, Backward uniqueness for parabolic equations, Arch. Ration. Mech. Anal., 169:147–157, 2003.
D. Fang and C. Qian, The regularity criterion for 3D Navier–Stokes equations involving one velocity gradient component, Nonlinear Anal., Theory Methods Appl., 78:86–103, 2013.
I. Kukavica and W.S. Ożański, An anisotropic regularity condition for the 3D incompressible Navier–Stokes equations for the entire exponent range, Appl. Math. Lett., 122:107298, 2021.
I. Kukavica, W. Rusin, and M. Ziane, Localized anisotropic regularity conditions for the Navier–Stokes equations, J. Nonlinear Sci., 27(6):1725–1742, 2017.
I. Kukavica and M. Ziane, Navier–Stokes equations with regularity in one direction, J. Math. Phys., 48(6):065203, 2007.
P. Kučera and Z. Skalák, Smoothness of the velocity time derivative in the vicinity of regular points of the Navier–Stokes equations, in K. Kozel, J. Přihoda, and M. Feistauer (Eds.), Proceedings of the 4th Seminar “Euler and Navier–Stokes equations (Theory, Numerical Solution, Applications)”, Institute of Thermomechanics AS CR, Praha, 2001, pp. 83–86.
P. Kučera and Z. Skalák, Regularity of suitable weak solutions of the Navier–Stokes equations as consequence of regularity of two velocity components, in K. Kozel and J. Přihoda (Eds.), Topical Problems of Fluid Mechanics 2002, Institute of Thermomechanics AS CR, Praha, 2002, pp. 53–56.
Y. Liu and P. Zhang, Critical one component anisotropic regularity for 3-D Navier–Stokes system, Sci. Sin., Math., 49(10):1405–1430, 2017.
Y. Namlyeyeva and Z. Skalak, The optimal regularity criterion for the Navier–Stokes equations in terms of one directional derivative of the velocity, ZAMM, Z. Angew. Math. Mech., 100(1):e201800114, 2019.
J. Neustupa, A. Novotný, and P. Penel, Interior regularity of a weak solution to the Navier–Stokes equations in dependence on one component of velocity, in G.P. Galdi and R. Rannacher (Eds.), Topics in mathematical fluid mechanics. Meeting on the occasion of Professor John G. Heywood sixtieth birthday, Capo Miseno, Italy, May 27–30, 2000, Quad. Mat., Vol. 10, Aracne, Rome, 2002, pp. 168–183.
J. Neustupa and P. Penel, Regularity of a suitable weak solution to the Navier–Stokes equations as a consequence of regularity of one velocity component, in A. Sequeira, H. Beirão da Vega, and J.H. Videman (Eds.), Applied Nonlinear Analysis. In honor of the 70th birthday of Professor Jindřich Nečas, Kluwer/Plenum, New York, 1999, pp. 391–402.
J. Neustupa and P. Penel, Anisotropic and geometric criteria for interior regularity of weak solutions to the 3D Navier–Stokes equations, in J. Neustupa and P. Penel (Eds.), Mathematical Fluid Mechanics: Recent Results and Open Problems, Adv. Math. Fluid Mech., Birkhäuser, Basel, 2001, pp. 237–268.
G. Prodi, Un teorema di unicita per el equazioni di Navier–Stokes, Ann. Mat. Pura Appl. (4), 48:173–182, 1959.
V. Scheffer, Partial regularity of solutions to the Navier–Stokes equations, Pacific J. Math., 66(2):535–552, 1976.
J. Serrin, The initial value problems for the Navier–Stokes equations, in R.E. Langer (Ed.), Nonlinear Problems. Univ. Wisconsin Press, Madison, 1963.
Z. Skalak, The end-point regularity criterion for the Navier–Stokes equations in terms of ∂3u, Nonliner Anal., Real World Appl., 55:103120, 2020.
Z. Zhang, A improved regularity criterion for the Navier–Stokes equations in terms of one directional derivative of the velocity field, Bull. Math. Sci., 8:33–47, 2018.
Z. Zhang, F. Alzahrani, T. Hayat, and Y. Zhou, Two new regularity criteria for the Navier–Stokes equations via two entries of the velocity Hessian tensor, Appl. Math. Lett., 37:124–130, 2014.
Z. Zhang, D. Zhong, and X. Huang, A refined regularity criterion for the Navier–Stokes equations involving one non-diagonal entry of the velocity gradient, J. Math. Anal. Appl., 453(2):1145–1150, 2017.
Y. Zhou and M. Pokorný, On a regularity criterion for the Navier–Stokes equations involving gradient of one velocity component, J. Math. Phys., 50(12):123514, 2009.
Y. Zhou and M. Pokorný, On the regularity of the solutions of the Navier–Stokes equations via one velocity component, Nonlinearity, 23(5):1097–1107, 2010.
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.
Zhengguang Guo was partially supported by NSFC under grant No. 11301394.
Petr Kucera and Zdenek Skalak were supported by the European Regional Development Fund, project No. CZ.02.1.01/0.0/0.0/16_019/0000778.
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
Guo, Z., Kucera, P. & Skalak, Z. The local regularity conditions for the Navier–Stokes equations via one directional derivative of the velocity. Lith Math J 62, 333–348 (2022). https://doi.org/10.1007/s10986-022-09573-w
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10986-022-09573-w