Advertisement

Nonhomogeneous boundary value problem for the stationary Navier–Stokes equations in a domain with a cusp

  • Kristina Kaulakytė
  • Neringa Klovienė
  • Konstantin PileckasEmail author
Article
  • 17 Downloads

Abstract

The nonhomogeneous boundary value problem for the stationary Navier–Stokes equations in a two-dimensional domain with a cusp point on the boundary is studied. The case when the flux of the boundary value \(\mathbf{a }\) is nonzero, i.e., when there is a source or sink in the cusp point, is considered. The existence of at least one weak solution having infinite Dirichlet integral is proved without any restrictions on the size of the flux \(F=\int \limits _{\partial \Omega }\mathbf{a }\cdot \mathbf{n }dS\).

Keywords

Stationary Navier–Stokes equations Nonhomogeneous boundary condition Cusp point singularity Nonzero flux 

Mathematics Subject Classification

35Q30 76D03 76D05 

Notes

Acknowledgements

The research was funded by a Grant No. S-MIP-17-68 from the Research Council of Lithuania.

References

  1. 1.
    Chipot, M., Kaulakyte, K., Pileckas, K., Xue, W.: On nonhomogeneous boundary value problems for the stationary Navier–Stokes equations in two dimensional symmetric semi-infinite outlets. Anal. Appl. 15(4), 543–569 (2017)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Eismontaite, A., Pileckas, K.: On singular solutions of time-periodic and steady Stokes problems in a power cusp domain. Appl. Anal. 97(3), 415–437 (2018)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Eismontaite, A., Pileckas, K. : On singular solutions of the initial boundary value problem for the Stokes system in a power cusp domain. Appl. Anal. (2018).  https://doi.org/10.1080/00036811.2018.1460815
  4. 4.
    Kim, H., Kozono, H.: Removable isolated singularity theorem for the stationary Navier–Stokes equations. J. Differ. Equ. 220, 68–84 (2006)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Korobkov, M.V., Pileckas, K., Pukhnache, V.V., Russo, R.: The flux problem for the Navier–Stokes equations. Uspech. Mat Nauk 69(6), 115–176 (2014). English Transl.: Russian Math Surv., 69, issue 6, (2014), 1065–1122MathSciNetCrossRefGoogle Scholar
  6. 6.
    Kozlov, V., Rossmann, J.: On the nonstationary Stokes system in a cone. J. Differ. Equ. 260(12), 8277–8315 (2016)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Ladyzhenskaya, O.A.: The Mathematical Theory of Viscous Incompressible Flow, 2nd edn. Gordon and Breach, London (1969)zbMATHGoogle Scholar
  8. 8.
    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.: J. Sov. Math.10, No. 2 (1978), 257-285MathSciNetzbMATHGoogle Scholar
  9. 9.
    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.: J. Sov. Math.21, No. 5 (1983), 728-761MathSciNetzbMATHGoogle Scholar
  10. 10.
    Nazarov, S.A.: On the flow of water under a still stone. Mat. Sbornik 11, 75–110 (1995). English Transl.: Math. Sbornik, 186, No. 11, (1995), 1621–1658zbMATHGoogle Scholar
  11. 11.
    Nazarov, S.A., Pileckas, K.: Asymptotics of solutions to Stokes and Navier–Stokes equations in domains with paraboloidal outlets to infinity. Rend. Sem. Math. Univ. Padova 99, 1–43 (1998)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Pukhnachev, V.V.: Singular solutions of Navier–Stokes equations. In: Advances in mathematical analysis of PDEs, Proceedings of the Saint Petersburg Mathematical Society XV, 232. English Transl.: AMS translated series, 2, American Mathematical Society, Providence, Rhode Island, (2014), 193–218Google Scholar
  13. 13.
    Russo, A., Tartaglione, A.: On the existence of singular solutions of the stationary Navier–Stokes problem. Lith. Math J. 54(4), 423–437 (2013)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Solonnikov, V.A., Pileckas, K.: Certain spaces of solenoidal vectors and the solvability of the boundary problem for the Navier–Stokes system of equations in domains with noncompact boundaries. Zapiski Nauchn. Sem. LOMI 73, 136–151 (1977). English Transl.: J. Sov. Math.34, No. 6 (1986), 2101-2111Google Scholar
  15. 15.
    Solonnikov, V.A.: Stokes and Navier-Stokes equations in domains with noncompact boundaries. In: Nonlinear Partial Differential Equations and Their Applications. Pitmann Notes in Math. College de France Seminar vol. 3, pp. 240–349 (1983)Google Scholar
  16. 16.
    Solonnikov, V.A.: On a boundary value problem for the Navier–Stokes equations with discontinuous boundary data. Rend. Mat. Ser. VII 10, 757–772 (1990)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Solonnikov, V.A.: On boundary value problem for the Navier–Stokes equations with discontinuous boundary data. In: Partial Differential Equations and Related Subjects. Pitman Research Notes in Math. Series vol. 282, pp. 86–97 (1993)Google Scholar
  18. 18.
    Solonnikov, V.A.: Free boundary problems for the Navier–Stokes equations with moving contacts points. In: Free Boundary Problems: Theory and Applications. Pitman Research Notes in Math. Series vol. 323, pp. 203–214 (1995)Google Scholar
  19. 19.
    Solonnikov, V.A.: On the problem of moving contact angle. In: Butazzo, G., Galdi, P., Lanconelli, E., Pucci, P. (eds.) Nonlinear Analysis and Continuum Mechanics, pp. 107–137. Springer, New York (1998)CrossRefGoogle Scholar
  20. 20.
    Solonnikov, V.A.: Solvability of two-dimensional free boundary problem for the Navier–Stokes equations for limiting values of contact angle. In: Recent Developments in Partial Differential Equations, Quaderni di Matematica vol. 2, pp. 163–210 (1998)Google Scholar
  21. 21.
    Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, New Jersey (1970)zbMATHGoogle Scholar
  22. 22.
    Videman, J.H., Nazarov, S.A., Sequeira, A.: Asymptotic modeling of a piston with completely wetted surface. Zapiski Nauchn. Sem. POMI 306, 53–70 (2003). English Transl.: Journal of Mathematical Sciences, 130, No. 4, (2005), 4803-4813zbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Kristina Kaulakytė
    • 1
  • Neringa Klovienė
    • 1
  • Konstantin Pileckas
    • 1
    Email author
  1. 1.Institute of Applied MathematicsVilnius UniversityVilniusLithuania

Personalised recommendations