Topological Methods in Nonlinear Analysis

On nonhomogeneous boundary value problem for the steady Navier-Stokes system in domain with paraboloidal and layer type outlets to infinity

Kristina Kaulakytė

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Abstract

The nonhomogeneous boundary value problem for the steady Navier-Stokes system is studied in a domain $\Omega$ with two layer type and one paraboloidal outlets to infinity. The boundary $\partial\Omega$ is multiply connected and consists of the outer boundary $S$ and the inner boundary $\Gamma$. The boundary value ${a}$ is assumed to have a compact support. The flux of ${a}$ over the inner boundary $\Gamma$ is supposed to be sufficiently small. We do not impose any restrictions on fluxes of ${a}$ over the unbounded components of the outer boundary $S$. The existence of at least one weak solution is proved.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 46, Number 2 (2015), 835-865.

Dates
First available in Project Euclid: 21 March 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1458588665

Digital Object Identifier
doi:10.12775/TMNA.2015.070

Mathematical Reviews number (MathSciNet)
MR3494974

Zentralblatt MATH identifier
1362.35211

Citation

Kaulakytė, Kristina. On nonhomogeneous boundary value problem for the steady Navier-Stokes system in domain with paraboloidal and layer type outlets to infinity. Topol. Methods Nonlinear Anal. 46 (2015), no. 2, 835--865. doi:10.12775/TMNA.2015.070. https://projecteuclid.org/euclid.tmna/1458588665


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References

  • Ch.J. Amick, Existence of solutions to the nonhomogeneous steady Navier–Stokes equations, Indiana Univ. Math. J. 33 (1984), 817–830.
  • W. Borchers and K. Pileckas, Note on the flux problem for stationary Navier–Stokes equations in domains with multiply connected boundary, Acta App. Math. 37 (1994), 21–30.
  • R. Farwig, H. Kozono and T. Yanagisawa, Leray's inequality in general multi-connected domains in ${\mathbb{R}}^n$, Math. Ann. 354 (2012), 137–145.
  • R. Farwig and H. Morimoto, Leray's inequality for fluid flow in symmetric multi-connected two-dimensional domains, Tokyo J. Math. 35, No. 1 (2012), 63–70.
  • R. Finn, On the steady-state solutions of the Navier–Stokes equations, III, Acta Math. 105 (1961), 197–244.
  • H. Fujita, On the existence and regularity of the steady-state solutions of the Navier–Stokes theorem, J. Fac. Sci. Univ. Tokyo Sect. I (1961) 9, 59–102.
  • H. Fujita, On stationary solutions to Navier–Stokes equation in symmetric plane domain under general outflow condition, Pitman research notes in mathematics, Proceedings of International conference on Navier–Stokes equations. Theory and numerical methods. June 1997. Varenna, Italy (1997) 388, 16-30.
  • H. Fujita and H. Morimoto, A remark on the existence of the Navier–Stokes flow with non-vanishing outflow condition, GAKUTO Internat. Ser. Math. Sci. Appl. 10 (1997), 53–61.
  • G.P. Galdi, On the existence of steady motions of a viscous flow with non–homogeneous conditions, Le Matematiche 66 (1991), 503–524.
  • J.G. Heywood, On uniqueness questions in the theory of viscous flow, Acta. Math. 136, (1976), 61–102.
  • ––––, On the impossibility, in some cases, of the Leray-Hopf condition for energy estimates, J. Math. Fluid Mech. 13, No. 3 (2011), 449–457.
  • E. Hopf, Ein allgemeiner Endlichkeitssats der Hydrodynamik, Math. Ann. 117 (1941), 764–775.
  • L.V. Kapitanskiĭ and K. Pileckas, 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), 5–36; English transl.: Proc. Math. Inst. Steklov 159 (1984), 3–34.
  • K. Kaulakyt\.e and K. Pileckas, On the nonhomogeneous boundary value problem for the Navier–Stokes system in a class of unbounded domains, J. Math. Fluid Mech., 14, No. 4 (2012), 693–716.
  • M.V. Korobkov, K. Pileckas and R. Russo, On the flux problem in the theory of steady Navier–Stokes equations with nonhomogeneous boundary conditions, Arch. Rational Mech. Anal. 207, No. 1 (2013), 185–213.
  • M.V. Korobkov, K. Pileckas and R. Russo, Steady Navier–Stokes system with nonhomogeneous boundary conditions in the axially symmetric case, C.R. Mecanique 340 (2012), 115–119.
  • ––––, Solution of Leray's problem for stationary Navier–Stokes equations in plane and axially symmetric spatial domains, arXiv:1302.0731, [math-ph], 4 Feb 2013.
  • H. Kozono and T. Yanagisawa, Leray's problem on the stationary Navier–Stokes equations with inhomogeneous boundary data, Math. Z. 262, No. 1 (2009), 27–39.
  • O.A. Ladyzhenskaya, Investigation of the Navier–Stokes equations in the case of stationary motion of an incompressible fluid, Uspech Mat. Nauk 3 (1959), 75–97 (in Russian).
  • ––––, The Mathematical Theory of Viscous Incompressible Fluid, Gordon and Breach (1969).
  • O.A. Ladyzhenskaya and V.A. Solonnikov, Some problems of vector analysis and generalized formulations of boundary value problems for the Navier–Stokes equations, Zap.Nauchn. Sem. LOMI 59 (1976), 81–116; English transl.: J. Sov. Math. 10, No. 2 (1978), 257–285.
  • ––––, 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) (1977), 39–47; English transl.: Vestnik Leningrad Univ. Math. 10 (1982), 271–280.
  • ––––, Determination of the solutions of boundary value problems for stationary Stokes and Navier–Stokes equations having an unbounded Dirichlet integral, Zap. Nauchn. Sem. LOMI 96 (1980), 117–160; English transl.: J. Sov. Math., 21, No. 5 (1983), 728–761.
  • J. Leray, Étude de diverses équations intégrales non linéaire et de quelques problèmes que pose l'hydrodynamique, J. Math. Pures Appl. 12 (1933), 1–82.
  • H. Morimoto, Stationary Navier–Stokes flow in \rom2D channels infolving the general outflow condition, Handbook of differential equations: stationary partial differential equations 4, Ch. 5, Elsevier (2007), 299–353.
  • ––––, A remark on the existence of \rom2D steady Navier–Stokes flow in bounded symmetric domain under general outflow condition, J. Math. Fluid Mech. 9, No. 3 (2007), 411–418.
  • H. Morimoto and H. Fujita, A remark on the existence of steady Navier–Stokes flows in \rom2D semi-infinite channel infolving the general outflow condition, Math. Bohem. 126, No. 2 (2001), 457–468.
  • ––––, A remark on the existence of steady Navier–Stokes flows in a certain two–dimensional infinite channel, Tokyo J. Math. 25, No. 2 (2002), 307–321.
  • ––––, Stationary Navier–Stokes flow in $2$-dimensional $Y$-shape channel under general outflow condition, The Navier–Stokes Equations: Theorey and Numerical Methods, Lecture Note in Pure and Applied Mathematics, Marcel Decker (Morimoto Hiroko, Other) 223, (2002), 65–72.
  • S.A. Nazarov and K. Pileckas, On the solvability of the Stokes and Navier–Stokes problems in domains that are layer-like at infinity, J. Math. Fluid Mech. 1, No. 1 (1999), 78–116.
  • J. Neustupa, On the steady Navier–Stokes boundary value problem in an unbounded $2D$ domain with arbitrary fluxes through the components of the boundary, Ann. Univ. Ferrara, 55, No. 2 (2009), 353–365.
  • ––––, A new approach to the existence of weak solutions of the steady Navier–Stokes system with inhomoheneous boundary data in domains with noncompact boundaries, Arch. Rational Mech. Anal 198, No. 1 (2010), 331–348.
  • K. Pileckas, Existence of solutions for the Navier–Stokes equations, having an infinite dissipation of energy in a class of domains with noncompact boundaries, Zap. Nauchn. Sem. LOMI 110, (1981), 180–202.
  • V.V. Pukhnachev, Viscous flows in domains with a multiply connected boundary, New Directions in Mathematical Fluid Mechanics. The Alexander V. Kazhikhov Memorial Volume (A.V. Fursikov, G.P. Galdi and V.V. Pukhnachev, eds.), Basel – Boston – Berlin, Birkhäuser (2009), 333–348.
  • ––––, The Leray problem and the Yudovich hypothesis, Izv. Vuzov. Sev.–Kavk. Region. Natural Sciences, the special issue “Actual problems of mathematical hydrodynamics” (2009), 185–194 (in Russian).
  • L.I. Sazonov, On the existence of a stationary symmetric solution of the two–dimensional fluid flow problem, Mat. Zametki 54, No. 6 (1993), 138–141; English transl.: Math. Notes 54, No. 6 (1993), 1280–1283.
  • V.A. Solonnikov, On the solvability of boundary and initial-boundary value problems for the Navier–Stokes system in domains with noncompact boundaries, Pacific J. Math. 93, No. 2 (1981), 443–458.
  • ––––, Stokes and Navier–Stokes equations in domains with noncompact boundaries, Nonlinear Partial Differential Equations and their Applications. Pitmann Notes in Math., College de France Seminar 3 (1983), 240–349.
  • ––––, On solutions of stationary Navier–Stokes equations with an infinite Dirichlet integral, Zap. Nauchn. Sem. LOMI 115 (1982), 257–263; English transl.: J. Sov. Math., 28, No. 5 (1985), 792–799.
  • ––––, Boundary and initial–boundary value problems for the Navier–Stokes equations in domains with noncompact boundaries, Math. Topics in Fluid Mechanics, Pitman Research Notes in Mathematics Series 274 (J.F. Rodriques and A. Sequeira, eds.) (1991), 117–162.
  • ––––, On problems for hydrodynamics of viscous flow in domains with noncompact boundaries, Algebra i Analiz 4, No. 6 (1992), 28–53; English transl.: St. Petersburg Math. J. 4, No. 6 (1992).
  • V.A. Solonnikov and K. Pileckas, 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, Zap. Nauchn. Sem. LOMI 73 (1977), 136–151; English transl.: J. Sov. Math. 34, No. 6 (1986), 2101–2111.
  • E.M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press (1970).
  • A. Takeshita, A remark on Leray's inequality, Pacific J. Math. 157 (1993), 151–158.
  • I.I. Vorovich and V.I. Judovich, Stationary flows of a viscous incompres-sible fluid, Mat. Sb. 53 (1961), 393–428 (in Russian).