SpringerLink
Log in

Existence, uniqueness and asymptotics of steady jets

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

Abstract

We consider the three-dimensional flow through an aperture in a plane either with a prescribed flux or pressure drop condition. We discuss the existence and uniqueness of solutions for small data in weighted spaces and derive their complete asymptotic behaviour at infinity. Moreover, we show that each solution with a bounded Dirichlet integral, which has a certain weak additional decay, behaves like O(r −2) as r=¦x¦→∞ and admits a wide jet region. These investigations are based on the solvability properties of the linear Stokes system in a half space ℝ 3+ . To investigate the Stokes problem in ℝ 3+ , we apply the Mellin transform technique and reduce the Stokes problem to the determination of the spectrum of the corresponding invariant Stokes-Beltrami operator on the hemisphere.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Agmon, S., Douglis, A. & Nirenberg, L., Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II. Comm. Pure Appl. Math. 17, 35–92 (1964).

    MathSciNet  Google Scholar 

  2. Adams, R. A., Sobolev spaces, Academic Press, New York, San Francisco, London 1975.

    Google Scholar 

  3. Agranovich, M. S. & Vishik, M. I., Elliptic problems with a parameter and parabolic problems of general type, Uspekhi Mat. Nauk, 19 (117) 53–161 (1964). English Transl. in Russian Math. Surveys 19, 53–158 (1964).

    Google Scholar 

  4. Babenko, K. I., On stationary solutions of the problem of flow past a body of a viscous incompressible fluid, Math. USSR Sbornik 20, 1–25 (1973).

    MATH  MathSciNet  Google Scholar 

  5. Blum, H. & Rannacher, R., On the boundary value problem of the biharmonic operator on domains with angular comers, Math. Meth. Appl. Sci. 2, 556–581 (1980).

    MathSciNet  Google Scholar 

  6. Blum, H., Lin, Q. & Rannacher, R., Asymptotic error expansion and Richardson extrapolation for linear finite elements, Numer. Math. 49, 11–37 (1986).

    Article  MathSciNet  Google Scholar 

  7. Borchers, W. & Miyakawa, T., Algebraic L 2 decay for Navier-Stokes flows in exterior domains, Acta. Math. 165, 189–227 (1990).

    MathSciNet  Google Scholar 

  8. Borchers, W. & Sohr, H., On the equations rot ν=g and div u=f with zero boundary conditions, Hokkaido Math. J. 19, 67–87 (1990).

    MathSciNet  Google Scholar 

  9. Borchers, W., Galdi G. P. & Pileckas, K., On the niqueness of Leray-Hopf solutions for the flow through an aperture, preprint.

  10. Cattabriga, L., Su un problema al contorno relativo al sistema di equazioni di Stokes, Rend. Sem. Math. Univ. Padova 31, 308–340 (1961).

    MATH  MathSciNet  Google Scholar 

  11. Céa, J., Optimisation théorie et algorithmes, Dunod, Paris, 1971.

    Google Scholar 

  12. Galdi, G. P., On the asymptotic properties of Leray's solutions to the exterior stationary three-dimensional Navier-Stokes equations, to appear in Degenerate Diffusion, edited by Ni, W.-M., Peletier, L. A. & Vazquez, J. L., Springer-Verlag.

  13. Gohberg, I. C. & Krein M. G., Introduction to the theory of linear non-selfadjoint operators. Amer. Math. Soc., Providence, 1969.

    Google Scholar 

  14. Grisvard, P., Elliptic problems in nonsmooth domains, Pitman, Boston, London, Melbourne, 1985.

    Google Scholar 

  15. Finn, R., On the exterior stationary problem for the Navier-Stokes equations and associated perturbation problems, Arch. Rational Mech. Anal. 19, 363–406 (1965).

    Article  MATH  MathSciNet  Google Scholar 

  16. Finn, R., On the steady-state solutions of the Navier-Stokes equations, III, Acta Math. 105, 197–244 (1961).

    MATH  MathSciNet  Google Scholar 

  17. Heywood, J. G., On uniqueness questions in the theory of viscous flow, Acta Math. 136, 61–102 (1976).

    MATH  MathSciNet  Google Scholar 

  18. Heywood, J. G., Open problems in the theory of the Navier-Stokes equations for viscous incompressible flow. Proceedings of the conference, The Navier-Stokes equations: Theory and Numerical Methods, Oberwolfach, 1988, Springer Lecture Notes in Math., 1431 (eds. Heywood, J. G., Masuda, K., Rautmann, R. & Solonnikov, V. A.) 1–22, 1990.

  19. Kapitanskii, L. V., Coincidence of the spaces \(\mathop {J_2^1 }\limits^ \circ (\Omega )\) and \(\hat J_2^1 (\Omega )\) for plane domians Ω having exists at infinity, Zapiski Nauch. Semin. LOMI, 110, 74–81 (1981). English Transl. in J. Sov. Math. 25, 850–855 (1984).

    MATH  MathSciNet  Google Scholar 

  20. Kapitanskii, L. V. & Pileckas, K., On spaces of solenoidal vector fields and boundary value problems for the Navier-Stokes equations in domains with noncompact boundaries, Trudy Mat. Inst. Steklov 159 (1983). English Transl. in Proc. Math. Inst. Steklov 159, 3–34 (1984).

  21. Kapitanskii, L. V. & Pileckas, K., Certain problems of vector analysis, Zapiski Nauchn. Sem. LOMI 138, 65–85 (1984). English Transl. in J. Sov. Math. 32, 469–483 (1986).

    MathSciNet  Google Scholar 

  22. Kellogg, R. B. & Osborn, J. E., A regularity result for the Stokes problem in a convex polygon, J. Funct. Analysis 21, 397–431 (1976).

    Article  MathSciNet  Google Scholar 

  23. Kondrat'ev, V. A., Boundary value problems for elliptic equations in domains with conical or corner points, Trudy Moskov. Mat Obshch. 16, 209–292 (1967). English transl. in Trans. Moscow Math. Soc. 16 (1967).

    MATH  Google Scholar 

  24. Ladyzhenskaya, O. A., The mathematical theory of viscous incompressible flow, Gordon and Breach, New York, London, Paris, 1969.

    Google Scholar 

  25. Ladyzhenskaya, O. A. & Solonnikov, V. A., Some problems of vector analysis and generalized formulations of boundary value problems for the Navier-Stokes equations, Zapiski Nauchn. Sem. LOMI 59, 81–116 (1976). English transl. in J. Sov. Math. 10, 257–285 (1978).

    Google Scholar 

  26. Ladyzhenskaya, O. A. & Solonnikov, V. A., On the solvability of boundary value problems for the Navier-Stokes equations in regions with noncompact boundaries, Vestnik Leningrad. Univ. 13 (Ser. Mat. Mekh. Astr. vyp. 3), 39–47 (1977). English transl. in Vestnik Leningrad Univ. Math. 10, 271–280 (1982).

    Google Scholar 

  27. Ladyzhenskaya, O. A. & Solonnikov, V. A., Determination of the solutions of boundary value problems for stationary Stokes and Navier-Stokes equations having an unbounded Dirichlet integral, Zapiski Nauchn. Sem. LOMI 96, 117–160 (1980). English transl. in J. Sov. Math. 21, 728–761 (1983).

    MathSciNet  Google Scholar 

  28. Landau, L. D. & Lifschitz, E. M., Hydromechanik, Akademie-Verlag, Berlin 1971.

    Google Scholar 

  29. Liusternik, L. A., On conditional extrema of functions, Math. USSR Sbornik 41 (1934).

  30. Maz'ya, V. G. & Plamenevskii, B. A., Estimates in L p and Hölder classes and the Miranda-Agmon maximum principle for solutions of elliptic boundary value problems in domains with singular points on the boundary, Math. Nachr. 81, 25–82 (1978). English transl. in Amer. Math. Soc. Transl. 123 (2) 1–56 (1984).

    MathSciNet  Google Scholar 

  31. Maz'ya, V. G. & Plamenevskii, B. A., The coefficients in the asymptotic expansion of the solutions of elliptic boundary value problems in a cone, Zapiski Nauchn. Sem. LOMI 52, 110–127 (1975). English transl. in J. Sov. Math. 9, 750–764 (1978).

    Google Scholar 

  32. Maz'ya, V. G. & Plamenevskii, B. A., On the coefficients in the asymptotics of solutions of elliptic boundary value problems in domains with conical points, Math. Nachr. 76, 29–60 (1977). English transl. in Amer. Math. Soc. Transl. 123 (2), 57–88 (1984).

    MathSciNet  Google Scholar 

  33. Morrey, C. B., Multiple integrals in the calculus of variations, Springer-Verlag, Berlin, Heidelberg, New York, 1966.

    Google Scholar 

  34. Pileckas, K., On unique solvability of boundary value problems for the Stokes system of equations in domains with noncompact boundaries, Trudy Mat. Inst. Steklov 147 (1980). English transl. in Proc. Math. Inst. Steklov 147, 117–126 (1981).

  35. Pileckas, K., Existence of solutions for the Navier-Stokes equations having an infinite dissipation of energy, in a class of domains with noncompact boundaries, Zapiski Nauchn. Sem. LOMI 110, 180–202 (1981). English transl. in J. Sov. Math. 25, 932–947 (1984).

    MATH  MathSciNet  Google Scholar 

  36. Pileckas, K., On spaces of solenoidal vectors, Trudy Mat. Inst. Steklov 159, (1983). English transl. in Proc. Math. Inst. Steklov 159, 141–154 (1984).

  37. Solonnikov, V. A., Stokes and Navier-Stokes equations in domains with noncompact boundaries, Collège de France Seminar 4, 240–349 (1983).

    MATH  MathSciNet  Google Scholar 

  38. Solonnikov, V. A., On the solvability of boundary and initial-boundary value problems for the Navier-Stokes system in domains with noncompact boundaries, Pacific J. Math. 93, 443–458 (1981).

    MATH  MathSciNet  Google Scholar 

  39. Solonnikov, V. A., On the Stokes equation in domains with nonsmooth boundaries and on a viscous incompressible flow with a free surface, Collège de France Seminar 3, 340–423, 1980/1981.

    Google Scholar 

  40. Soionnikov, V. A. & Shchadilov, V. E., On a boundary value problem for a stationary system of Navier-Stokes equations, Trudy Mat. Inst. Steklov 125, 196–210 (1973). English transl. in Proc. Mat. Inst. Steklov 125, 186–199 (1973).

    MathSciNet  Google Scholar 

  41. Solonnikov, V. A. & Pileckas, K., Certain spaces of solenoidal vectors and the solvability of the boundary value problem for the Navier-Stokes system of equations in domains with noncompact boundaries, Zapiski Nauchn. Sem. LOMI 73, 136–151 (1977). English transl. in J. Sov. Math. 34, 2101–2111 (1986).

    MathSciNet  Google Scholar 

  42. Widder, D. V., The Laplace transform, Princeton University Press 1941.

  43. Kozlov, V. A., Maz'ya, V. G. & Schwab, C., On singularities of solutions to the Dirichlet problem of hydrodynamics near the vertex of a 3-d cone, Report No. LITHMAT-R-9023, Linkö** (1990).

  44. Chang, H., The steady Navier-Stokes problem for low Reynolds number viscous jets, preprint.

Download references

Author information

Authors and Affiliations

  1. Fachbereich Mathematik-Informatik der Universität-GHS Paderborn, Warburgerstr. 100, W-4790, Paderborn, Germany

    Wolfgang Borchers & Konstantin Pileckas

Authors
  1. Wolfgang Borchers
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Konstantin Pileckas
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by J. Serrin

Rights and permissions

Reprints and permissions

About this article

Cite this article

Borchers, W., Pileckas, K. Existence, uniqueness and asymptotics of steady jets. Arch. Rational Mech. Anal. 120, 1–49 (1992). https://doi.org/10.1007/BF00381276

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00381276

Keywords

Search

Navigation