Mathematische Annalen

, Volume 374, Issue 3–4, pp 1559–1596 | Cite as

On a two components condition for regularity of the 3D Navier–Stokes equations under physical slip boundary conditions on non-flat boundaries

  • Hugo Beirão da Veiga
  • Josef BemelmansEmail author
  • Johannes Brand


This work concerns the sufficient condition for the regularity of solutions to the evolution Navier–Stokes equations known in the literature as Prodi–Serrin condition. H.-O. Bae and H. J. Choe proved in 1997 that, in the whole space \(\mathbb {R}^3,\) it is sufficient that two components of the velocity satisfy the above condition in order to guarantee the regularity of solutions. In 2017, H. Beirão da Veiga extended this result (Beirão da Veiga, J Math Anal Appl 453:212–220, 2017) to the half-space case \(\mathbb {R}^n_+\) under slip boundary conditions by assuming that the velocity components parallel to the boundary enjoy the above condition. It remained open whether the flat boundary geometry is essential. Below, we prove that, under physical slip boundary conditions imposed in cylindrical boundaries, the result still holds.


Navier–Stokes Equations Slip Boundary Conditions Prodi–Serrin Condition Two Components Condition Regularity 

Mathematics Subject Classification



  1. 1.
    Bae, H.-O., Choe, H.J.: A regularity criterion for the Navier–Stokes equations. Commun. Partial Differ. Equations 32, 1173–1187 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bae, H.-O., Wolf, J.: A local regularity condition involving two velocity components of Serrin-type for the Navier–Stokes equations. C. R. Acad. Sci. Paris Ser. I(354), 167–174 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Batchelor, G.K.: An Introduction to Fluid Dynamics. Cambridge University Press, Cambridge (1967)zbMATHGoogle Scholar
  4. 4.
    Beirão da Veiga, H.: Remarks on the smoothness of the \(L^\infty (0, T;L^3)\) solutions of the 3-D Navier–Stokes equations. Port. Math. 54, 381–391 (1997)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Beirão da Veiga, H.: On the smoothness of a class of weak solutions to the Navier–Stokes equations. J. Math. Fluid Mech. 2, 315–323 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Beirão da Veiga, H.: Regularity for Stokes and generalized Stokes systems under nonhomogeneous slip-type boundary conditions. Adv. Differ. Equations 9, 1079–1114 (2004)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Beirão da Veiga, H.: Remarks on the Navier–Stokes equations under slip type boundary conditions with linear friction. Port. Math. 64, 377–387 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Beirão da Veiga, H.: On the extension to slip boundary conditions of a Bae and Choe regularity criterion for the Navier-Stokes equations. The half space case. J. Math. Anal. Appl. 453, 212–220 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Berselli, L.C.: A note on regularity of weak solutions of the Navier-Stokes equations in \(R^n\). Jpn. J. Math. 28, 51–60 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Cao, C., Titi, E.S.: Regularity criteria for the three-dimensional Navier–Stokes equations. Indiana Univ. Math. J. 57, 2643–2661 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Chae, D., Choe, H.-J.: Regularity of solutions to the Navier–Stokes equation. Electron. J. Differ. Equations 05, 1–7 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Escauriaza, L., Seregin, G., Šverák, V.: \(L_{3,\infty }\)-Solutions to the Navier–Stokes equations and backward uniqueness. Russ. Math. Surv. 58, 211–250 (2003)CrossRefzbMATHGoogle Scholar
  13. 13.
    Foias, C.: Une remarque sur l’unicité des solutions des équations de Navier-Stokes en dimension \(n.\). Bull. Soc. Math. Fr. 89, 1–8 (1961)zbMATHGoogle Scholar
  14. 14.
    Galdi, G.P.: An Introduction to the Navier–Stokes initial-boundary value problems. In: Galdi, G.P., Heywood, M.I., Rannacher, R. (eds.) Fundamental Directions in Mathematical Fluid Mechanics. Advances in Mathematical Fluid Mechanics, pp. 1–70. Birkhäuser, Basel (2000)Google Scholar
  15. 15.
    Galdi, G.P., Maremonti, P.: Sulla regolarità delle soluzioni deboli al sistema di Navier–Stokes in domini arbitrari. Ann. Univ. Ferrara Sez. VII. Sci. Mat. 34, 59–73 (1988)Google Scholar
  16. 16.
    Giga, Y.: Solutions for semilinear parabolic equations in \(L^p\) and regularity of weak solutions of the Navier–Stokes system. J. Differ. Equations 62, 186–212 (1986)CrossRefzbMATHGoogle Scholar
  17. 17.
    He, C.: Regularity for solutions to the Navier–Stokes equations with one velocity component regular. Electron. J. Differ. Equations 29, 1–13 (2002)MathSciNetGoogle Scholar
  18. 18.
    Hopf, E.: Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen. Math. Nachr. 4, 213–231 (1951)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Kiselev, A.A., Ladyzhenskaya, O.A.: On the existence and uniqueness of the solution of the nonstationary problem for a viscous, incompressible fluid. Izv. Akad. Nauk SSSR Ser. Mat. 21, 655–680 (1957)MathSciNetGoogle Scholar
  20. 20.
    Kozono, H., Sohr, H.: Regularity criterion on weak solutions to the Navier–Stokes equations. Adv. Differ. Equations 2, 2924–2935 (2007)MathSciNetGoogle Scholar
  21. 21.
    Kukavica, I., Ziane, M.: Navier–Stokes equations with regularity in one direction. J. Math. Phys. 48, 2643–2661 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Ladyzhenskaya, O.A.: On uniqueness and smoothness of generalized solutions to the Navier–Stokes equations. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 5, 169–185 (1967)Google Scholar
  23. 23.
    Ladyzhenskaya, O.A.: La théorie mathématique des fluides visqueux incompressibles. Moscou (1961) [English edition. 2nd edn. Gordon & Breach, New York (1969)]Google Scholar
  24. 24.
    Leray, J.: Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math. 63, 193–248 (1934)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Lions, J.L.: Sur l’existence de solutions des équations de Navier–Stokes. C. R. Acad. Sci. Paris 248, 2847–2849 (1959)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Mikhailov, A.S., Shilkin, T.N.: \(\,L_{3,\,\infty }\)-solutions to the 3D-Navier–Stokes system in a domain with a curved boundary. J. Math. Sci. (N. Y.) 143, 2924–2935 (2007)Google Scholar
  27. 27.
    Neustupa, J., Penel, P.: Anisotropic and geometric criteria for interior regularity of weak solutions to the 3D Navier–Stokes equations. In: Neustupa, J., Penel, P. (eds.) Mathematical Fluid Mechanics. Advances in Mathematical Fluid Mechanics, pp. 237–265. Birkhäuser, Basel (2001)Google Scholar
  28. 28.
    Prodi, G.: Un teorema di unicità per le equazioni di Navier–Stokes. Ann. Mat. Pura Appl. 48, 173–182 (1959)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Prodi, G.: Résultats récents et problèmes anciens dans la théorie des équations de Navier-Stokes. In: Les Équations aux Dérivées Partielles, pp. 181–196. Éditions du CNRS, Paris (1962)Google Scholar
  30. 30.
    Serrin, J.: On the interior regularity of weak solutions of the Navier–Stokes equations. Arch. Ration. Mech. Anal. 9, 187–195 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Serrin, J.: The initial value problem for the Navier–Stokes equations. In: Langer, R.E. (ed.) Nonlinear Problems, pp. 69–98. University of Wisconsin Press, Madison (1963)Google Scholar
  32. 32.
    Sohr, H.: Zur Regularitätstheorie der instationären Gleichungen von Navier–Stokes. Math. Z. 184, 359–375 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Solonnikov, V.A., Ščadilov, V.E.: On a boundary value problem for a stationary system of Navier–Stokes equations. Proc. Steklov Inst. Math. 125, 186–199 (1973)MathSciNetzbMATHGoogle Scholar
  34. 34.
    von Wahl, W.: Regularity of weak solutions of the Navier–Stokes equations. Proc. Symp. Pure Math. 45, 497–503 (1986)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Zhang, Z., Zhong, D., Huang, X.: A refined regularity criterion for the Navier–Stokes equations involving one non-diagonal entry of the velocity gradient. J. Math. Anal. Appl. 453, 1145–1150 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Zhou, Y., Pokorný, M.: On the regularity of the solutions of the Navier–Stokes equations via one velocity component. Nonlinearity 23, 1097–1107 (2010)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsPisa UniversityPisaItaly
  2. 2.Institute for MathematicsRWTH Aachen UniversityAachenGermany
  3. 3.WZLRWTH Aachen UniversityAachenGermany

Personalised recommendations