Solvability in a finite pipe of steady-state Navier–Stokes equations with boundary conditions involving Bernoulli pressure

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Most methods of numerical simulation require a truncation of an infinite domain to a bounded one, thereby introducing artificial boundaries. We prove existence of weak solutions to the stationary Navier–Stokes equations, simulating the steady flow of a viscous fluid through the pipe. We consider the case when at inflow and outflow boundaries conditions involving the Bernoulli pressure are prescribed and study the problems either with given flow rate of the fluid or the given pressure drop. In both cases we prove existence of a weak solution without any restriction on the data.

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Fig. 1
Fig. 2


  1. 1.

    Without loss of generality we have assumed that the viscosity coefficient \(\nu \) and the density \(\rho \) are constant: \(\nu =1, \rho =1\).

  2. 2.

    with the boundary condition \(\partial _\mathbf{n} \mathbf{u}_{\varvec{\tau }}|_{\sigma _\pm }=0\) instead of \( \mathbf{u}_{\varvec{\tau }}|_{\sigma _\pm }=0\).

  3. 3.

    Recall that for the problem (1.6) the pressure drop \(p_*=p_+-p_-\) is given.


  1. 1.

    Amick, Ch.J.: Existence of solutions to the nonhomogeneous steady Navier-Stokes equations. Indiana Univ. Math. J. 33, 817–830 (1984)

  2. 2.

    Baffico, L., Grandmont, C., Maury, B.: Multiscale modeling of the respiratory tract. Math. Models Methods Appl. Sci. 20(1), 59–93 (2010)

  3. 3.

    Beneš, M., Kučera, P.: Solutions of the Navier–Stokes equations with various types of boundary conditions. Arch. Math. 98, 487–497 (2012)

  4. 4.

    Blazy, S., Nazarov, S., Specovius-Neugebauer, M.: Artificial boundary conditions of pressure type for viscous flows in a system of pipes. J. Math. Fluid Mech. 9(1), 1–33 (2007)

  5. 5.

    Braack, M., Mucha, P.P.: Directional do-nothing condition for the Navier–Stokes equations. J. Comput. Math. 32(5), 507–521 (2014)

  6. 6.

    Conca, C., Murat, F., Pironneau, O.: The Stokes and Navier–Stokes equations with boundary conditions involving the pressure. Jpn. J. Math. 20(2), 279–318 (1994)

  7. 7.

    Conca, C., Pares, C., Pironneau, O., Thiriet, M.: Navier–Stokes equations with imposed pressure and velocity fluxes. Int. J. Numer. Methods Fluids 20(4), 267–287 (1995)

  8. 8.

    Egloffe, A.-C.: Study of some inverse problems for the Stokes system. Application to the lungs, Thèse de doctorat, l’Université Pierre et Marie Curie—Paris VI (2012)

  9. 9.

    Fouchet-Incaux, J.: Artificial boundaries and formulations for the incompressible Navier–Stokes equations: applications to air and blood flows. SeMA 64, 1–40 (2014)

  10. 10.

    Galdi, G.P.: An Introduction to the Mathematical Theory of the Navier–Stokes Equation. Steady-State Problems, 2nd edn. Springer, Berlin (2011)

  11. 11.

    Galdi, G.P., Rannacher, R., Robertson, A.M., Turek, S.: Modeling, Analysis and Simulations, Oberwolfach Seminars. Birkhäuser, Basel (2008)

  12. 12.

    Grandmont, C., Maury, B., Soualah, A.: Multiscale modeling of the respiratory track: a theoretical framework. In: Mathematical and Numerical Modelling of the Human Lung, ESAIM Proceedings, vol. 23, pp. 10–29 (2008)

  13. 13.

    Gresho, P.M.: Some current CFD issues relevant to the incompressible Navier–Stokes equations. Comput. Methods Appl. Mech. Eng. 87, 201–252 (1991)

  14. 14.

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

  15. 15.

    Heywood, J.G., Rannacher, R., Turek, S.: Artificial boundaries and flux and pressure conditions for the incompressible Navier–Stokes equations. Int. J. Numer. Methods Fluids 22, 325–352 (1996)

  16. 16.

    Kapitanskii, L.V.: Stationary solutions of the Navier–Stokes equations in periodic tubes. Zapiski Nauch. Semin. LOMI 115, 104–113 (1982). (English Transl.: J. Sov. Math., 28 (1983) 689–695)

  17. 17.

    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, 5–36 (1983). (English Transl.: Proc. Math. Inst. Steklov, 159(2) (1983), 3–34)

  18. 18.

    Korobkov, M.V.: On Bernoulli law under minimal smoothness assumptions. Dokl. Math. 83, 107–110 (2011)

  19. 19.

    Korobkov, M.V., Pileckas, K., Russo, R.: On the flux problem in the theory of steady Navier–Stokes equations with nonhomogeneous boundary conditions. Arch. Ration. Mech. Anal. 207(1), 185–213 (2013)

  20. 20.

    Korobkov, M.V., Pileckas, K., Russo, R.: Steady Navier–Stokes system with nonhomogeneous boundary conditions in the axially symmetric case. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 14(1), 233–262 (2015)

  21. 21.

    Korobkov, M.V., Pileckas, K., Russo, R.: Solution of Leray’s problem for stationary Navier–Stokes equations in plane and axially symmetric spatial domains. Ann. Math. 181(2), 769–807 (2015)

  22. 22.

    Korobkov, M.V., Pileckas, K., Russo, R., Pukhnachev, V.V.: The flux problem for the Navier–Stokes equations. Russ. Math. Surv. 69(6), 1065–1122 (2014)

  23. 23.

    Kračmar, S., Neustupa, J.: Modeling of the unsteady flow through a channel with an artificial outflow condition by the Navier–Stokes variational inequality. Math. Nachr. 291, 1801–1814 (2018)

  24. 24.

    Kučera, P., Skalak, Z.: Solutions to the Navier–Stokes equations with mixed boundary conditions. Acta Appl. Math. 54(3), 275–288 (1998)

  25. 25.

    Ladyzhenskaya, O.A.: The Mathematical Theory of Viscous Incompressible Flow. Gordon and Breach, New York City (1969)

  26. 26.

    Ladyzhenskaya, O.A., Solonnikov, V.A.: On 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(2) (1978), 257–285)

  27. 27.

    Leray, J.: Étude de diverses équations intégrales non linéaire et de quelques problèmes que pose l’hydrodynamique. J. Math. Pures Appl. 12, 1–82 (1933)

  28. 28.

    Quarteroni, A., Veneziani, A.: Analysis of a geometrical multiscale model based on the coupling of ODEs and PDEs for blood flow simulations. Multiscale Model. Simul. 1(2), 173–195 (2003). (electronic)

  29. 29.

    Pileckas, K.: Navier–Stokes system in domains with cylindrical outlets to infinity. Leray’s Problem. Handbook of Mathematical Fluid Mechanics, Ch. 4, vol. 4, pp. 445–647. North-Holland Elsevier Sciences (2007)

  30. 30.

    Sani, R.L., Gresho, P.M.: Résumé and remarks on the open boundary condition minisymposium. Int. J. Numer. Methods Fluid 18, 983–1008 (1994)

  31. 31.

    Sani, R.L., Shen, J., Pironneau, O., Gresho, P.M.: Pressure boundary condition for the time-dependent incompressible Navier–Stokes equations. Int. J. Numer. Methods Fluids 50, 673–682 (2006)

  32. 32.

    Sazonov, L.I.: On the existence of a stationary symmetric solution of the two-dimensional fluid flow problem. Mat. Zametki 54(6), 138–141 (1993). (in Russian). English Transl.: Math. Notes, 54, No. 6 (1993), 1280–1283

  33. 33.

    Solonnikov, V.A.: On the solvability of boundary and initial-boundary value problems for the Navier–Stokes system in domains with noncompact boundaries. Pac. J. Math. 93(2), 443–458 (1981)

  34. 34.

    Specovius-Neugebauer, M.: Approximation of the Stokes Dirichlet problem in domains with cylindrical outlets. SIAM J. Math. Anal. 30(3), 645–677 (1999)

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M. Korobkov was partially supported by the Ministry of Education and Science of the Russian Federation (Grant 14.Z50.31.0037). The research of K. Pileckas was funded by the European Social Fund according to the activity “Improvement of researchers qualification by implementing world-class R and D projects” of Measure No. 09.3.3-LMT-K- 712.

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Correspondence to Konstantin Pileckas.

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Korobkov, M.V., Pileckas, K. & Russo, R. Solvability in a finite pipe of steady-state Navier–Stokes equations with boundary conditions involving Bernoulli pressure. Calc. Var. 59, 32 (2020).

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Mathematics Subject Classification

  • Primary 76D05
  • 35Q30
  • Secondary 31B10
  • 76D03