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, Volume 160, Issue 3–4, pp 359–383 | Cite as

The Navier–Stokes equations with the Neumann boundary condition in an infinite cylinder



We prove unique existence of local-in-time smooth solutions of the Navier–Stokes equations for initial data in \(L^{p}\) and \(p\in [3,\infty )\) in an infinite cylinder, subject to the Neumann boundary condition.

Mathematics Subject Classification

35Q35 35K90 


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The author is partially supported by JSPS through the Grant-in-aid for Young Scientist (B) 17K14217, Scientific Research (B) 17H02853 and Osaka City University Strategic Research Grant 2018 for young researchers.


  1. 1.
    Abe, K.: Vanishing viscosity limits for axisymmetric flows with boundary. arXiv:1806.04811
  2. 2.
    Adams, R.A., Fournier, J.J.F.: Sobolev Spaces. Volume 140 of Pure and Applied Mathematics (Amsterdam), 2nd edn. Elsevier/Academic Press, Amsterdam (2003)Google Scholar
  3. 3.
    Agmon, S., Douglis, A., Nirenberg, L.: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I. Commun. Pure Appl. Math. 12, 623–727 (1959)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Akiyama, T., Kasai, H., Shibata, Y., Tsutsumi, M.: On a resolvent estimate of a system of Laplace operators with perfect wall condition. Funkc. Ekvac. 47, 361–394 (2004)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bardos, C.: Existence et unicité de la solution de l’équation d’Euler en dimension deux. J. Math. Anal. Appl. 40, 769–790 (1972)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bergh, J., Löfström, J.: Interpolation spaces. An introduction. Springer, Berlin, New York (1976)CrossRefGoogle Scholar
  7. 7.
    Bourguignon, J.P., Brezis, H.: Remarks on the Euler equation. J. Funct. Anal. 15, 341–363 (1974)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Cattabriga, L.: Su un problema al contorno relativo al sistema di equazioni di Stokes. Rend. Sem. Mat. Univ. Padova 31, 308–340 (1961)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Clément, P., Prüss, J.: An operator-valued transference principle and maximal regularity on vector-valued \(L_p\)-spaces. In: Evolution Equations and Their Applications in Physical and Life Sciences (Bad Herrenalb, 1998), pp 67–87. Dekker, New York (2001)CrossRefGoogle Scholar
  10. 10.
    Denk, R., Hieber, M., Prüss, J.: \(R\)-boundedness, Fourier multipliers and problems of elliptic and parabolic type. Mem. Am. Math. Soc. 166(788), viii+114 (2003)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Duong, X.T.: \(H_\infty \) functional calculus of elliptic operators with \(C^\infty \) coefficients on \(L^p\) spaces of smooth domains. J. Aust. Math. Soc. Ser. A 48, 113–123 (1990)CrossRefGoogle Scholar
  12. 12.
    Ebin, D.G., Marsden, J.: Groups of diffeomorphisms and the motion of an incompressible fluid. Ann. Math. 2(92), 102–163 (1970)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Evans, L.C.: Partial Differential Equations, Volume 19 of Graduate Studies in Mathematics, 2nd edn. American Mathematical Society, Providence (2010)Google Scholar
  14. 14.
    Fabes, E.B., Lewis, J.E., Rivière, N.M.: Boundary value problems for the Navier–Stokes equations. Am. J. Math. 99, 626–668 (1977)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Farwig, R., Ri, M.-H.: An \(L^q(L^2)\)-theory of the generalized Stokes resolvent system in infinite cylinders. Stud. Math. 178(3), 197–216 (2007)CrossRefGoogle Scholar
  16. 16.
    Farwig, R., Ri, M.-H.: The resolvent problem and \(H^\infty \)-calculus of the Stokes operator in unbounded cylinders with several exits to infinity. J. Evol. Equ. 7, 497–528 (2007)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Farwig, R., Ri, M.-H.: Stokes resolvent systems in an infinite cylinder. Math. Nachr. 280, 1061–1082 (2007)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Farwig, R., Ri, M.-H.: Resolvent estimates and maximal regularity in weighted \(L^q\)-spaces of the Stokes operator in an infinite cylinder. J. Math. Fluid Mech. 10, 352–387 (2008)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Galdi, G.P.: An Introduction to the Mathematical Theory of the Navier–Stokes equations. Springer Monographs in Mathematics, 2nd edn. Springer, New York (2011)zbMATHGoogle Scholar
  20. 20.
    Geissert, M., Heck, H., Trunk, C.: \(H^\infty \)-calculus for a system of Laplace operators with mixed order boundary conditions. Discrete Contin. Dyn. Syst. Ser. S 6, 1259–1275 (2013)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Giga, Y., Miyakawa, T.: Solutions in \(L_r\) of the Navier–Stokes initial value problem. Arch. Ration. Mech. Anal. 89, 267–281 (1985)CrossRefGoogle Scholar
  22. 22.
    Kato, T.: Nonstationary flows of viscous and ideal fluids in \({ R}^{3}\). J. Funct. Anal. 9, 296–305 (1972)CrossRefGoogle Scholar
  23. 23.
    Kato, T., Lai, C.Y.: Nonlinear evolution equations and the Euler flow. J. Funct. Anal. 56, 15–28 (1984)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Lions, J.-L., Magenes, E.: Non-homogeneous Boundary Value Problems and Applications. Springer, New York, Heidelberg (1972)CrossRefGoogle Scholar
  25. 25.
    Lunardi, A.: Analytic semigroups and optimal regularity in parabolic problems. In: Brezis, H. (ed.) Progress in Nonlinear Differential Equations and their Applications, 16. Birkhäuser, Basel (1995)Google Scholar
  26. 26.
    McIntosh, A.: Operators which have an \(H_\infty \) functional calculus. In: Miniconference on Operator Theory and Partial Differential Equations (North Ryde, 1986), Volume 14 of Proceedings of the Centre Mathematical Analysis Australian National University, pp. 210–231. Australian National University, Canberra (1986)Google Scholar
  27. 27.
    Miyakawa, T.: The \(L^{p}\) approach to the Navier–Stokes equations with the Neumann boundary condition. Hiroshima Math. J. 10, 517–537 (1980)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Miyakawa, T.: On the initial value problem for the Navier–Stokes equations in \(L^{p}\) spaces. Hiroshima Math. J. 11, 9–20 (1981)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Miyakawa, T., Yamada, M.: Planar Navier–Stokes flows in a bounded domain with measures as initial vorticities. Hiroshima Math. J. 22, 401–420 (1992)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Muramatu, T.: On Besov spaces and Sobolev spaces of generalized functions definded on a general region. Publ. Res. Inst. Math. Sci. 9, 325–396 (1974)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Seeley, R.: Norms and domains of the complex powers \(A_{B}z\). Am. J. Math. 93, 299–309 (1971)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Sohr, H.: The Navier–Stokes equations. Birkhäuser Advanced Texts: Basler Lehrbücher. Birkhäuser Verlag, Basel (2001)CrossRefGoogle Scholar
  33. 33.
    Sohr, H., Thäter, G.: Imaginary powers of second order differential operators and \(L^q\)-Helmholtz decomposition in the infinite cylinder. Math. Ann. 311, 577–602 (1998)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series, No. 30. Princeton University Press, Princeton (1970)zbMATHGoogle Scholar
  35. 35.
    Swann, H.S.G.: The convergence with vanishing viscosity of nonstationary Navier–Stokes flow to ideal flow in \(R_{3}\). Trans. Am. Math. Soc. 157, 373–397 (1971)zbMATHGoogle Scholar
  36. 36.
    Tanabe, H.: Equations of Evolution, Volume 6 of Monographs and Studies in Mathematics. Pitman (Advanced Publishing Program), Boston (1979)Google Scholar
  37. 37.
    Temam, R.: On the Euler equations of incompressible perfect fluids. J. Funct. Anal. 20, 32–43 (1975)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Vishik, M.I., Komech, A.I.: Individual and statistical solutions of a two-dimensional Euler system. Dokl. Akad. Nauk SSSR 261, 780–785 (1981)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Weis, L.: Operator-valued Fourier multiplier theorems and maximal \(L_p\)-regularity. Math. Ann. 319, 735–758 (2001)MathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Department of Mathematics, Graduate School of ScienceOsaka City UniversityOsakaJapan

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