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Uspekhi Mat. Nauk, 2014, Volume 69, Issue 6(420), Pages 115–176 (Mi umn9616)  

This article is cited in 19 scientific papers (total in 19 papers)

The flux problem for the Navier–Stokes equations

M. V. Korobkova, K. Pileckasb, V. V. Pukhnachovcd, R. Russoe

a Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences
b Vilnius University, Vilnius, Lithuania
c Lavrentyev Institute of Hydrodynamics of Siberian Branch of the Russian Academy of Sciences
d Novosibirsk State University
e Seconda Università degli Studi di Napoli, Napoli, Italy

Abstract: This is a survey of results on the Leray problem (1933) for the Navier–Stokes equations of an incompressible fluid in a domain with multiple boundary components. Imposed on the boundary of the domain are inhomogeneous boundary conditions which satisfy the necessary requirement of zero total flux. The authors have proved that the problem is solvable in arbitrary bounded planar or axially symmetric domains. The proof uses Bernoulli's law for weak solutions of the Euler equations and a generalization of the Morse–Sard theorem for functions in Sobolev spaces. New a priori bounds for the Dirichlet integral of the velocity vector field in symmetric flows, as well as estimates for the regular component of the velocity in flows with singularities of source/sink type are presented.
Bibliography: 60 titles.

Keywords: Navier–Stokes and Euler equations, multiple boundary components, Dirichlet integral, virtual drain, Bernoulli's law, maximum principle.

Funding Agency Grant Number
Russian Foundation for Basic Research 14-01-00768-a
Ministry of Education and Science of the Russian Federation МД-5146.2013.1
Ministry of Health of the Republic of Lithuania CH-SMM-01/01
Siberian Branch of Russian Academy of Sciences 38
The first author was supported by grant no. 14-01-00768-a from the Russian Foundation for Basic Research and grant no. МД-5146.2013.1 for Support of Young Doctors of the Sciences from the President of the Russian Federation, the second author was supported by grant no. CH-ŠMM-01/01 from the Lithuanian–Swiss Cooperation Programme, and the third author was supported by the Siberian Branch of the Russian Academy of Sciences (grant no. 38 of the Programme of Joint Integration Projects of the Siberian, Ural, and Far-Eastern Branches of the Russian Academy of Sciences).


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English version:
Russian Mathematical Surveys, 2014, 69:6, 1065–1122

Bibliographic databases:

UDC: 517.59
MSC: Primary 35Q30, 35Q31, 76D05; Secondary 76D07, 76D10
Received: 20.08.2014

Citation: M. V. Korobkov, K. Pileckas, V. V. Pukhnachov, R. Russo, “The flux problem for the Navier–Stokes equations”, Uspekhi Mat. Nauk, 69:6(420) (2014), 115–176; Russian Math. Surveys, 69:6 (2014), 1065–1122

Citation in format AMSBIB
\by M.~V.~Korobkov, K.~Pileckas, V.~V.~Pukhnachov, R.~Russo
\paper The flux problem for the Navier--Stokes equations
\jour Uspekhi Mat. Nauk
\yr 2014
\vol 69
\issue 6(420)
\pages 115--176
\jour Russian Math. Surveys
\yr 2014
\vol 69
\issue 6
\pages 1065--1122

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    1. Baranovskii E.S., “A Novel 3D Model For Non-Newtonian Fluid Flows in a Pipe Network”, Math. Meth. Appl. Sci.  crossref  isi
    2. V. V. Pukhnachev, “Trekhmernaya simmetrichnaya zadacha protekaniya dlya uravnenii Nave–Stoksa”, Vestn. YuUrGU. Ser. Matem. modelirovanie i programmirovanie, 8:2 (2015), 95–104  mathnet  crossref  elib
    3. V. V. Pukhnachov, “Dirichlet Problem for the Stokes Equation”, Math. Notes, 101:1 (2017), 132–136  mathnet  crossref  crossref  mathscinet  isi  elib
    4. A. B. Morgulis, “Variatsionnye printsipy i ustoichivost otkrytykh techenii idealnoi neszhimaemoi zhidkosti”, Sib. elektron. matem. izv., 14 (2017), 218–251  mathnet  crossref
    5. V. Pukhnachev, “Symmetric solutions to the Leray problem”, C. R. Math. Acad. Sci. Paris, 355:1 (2017), 113–117  crossref  mathscinet  zmath  isi  scopus
    6. K. Ilin, A. Morgulis, “Inviscid instability of an incompressible flow between rotating porous cylinders to three-dimensional perturbations”, Eur. J. Mech. B Fluids, 61:1 (2017), 46–60  crossref  mathscinet  isi  elib  scopus
    7. A. Eismontaite, K. Pileckas, “On singular solutions of time-periodic and steady Stokes problems in a power cusp domain”, Appl. Anal., 97:3 (2018), 415–437  crossref  mathscinet  isi  scopus
    8. R. Juodagalvyte, K. Kaulakyte, “Time periodic boundary value Stokes problem in a domain with an outlet to infinity”, Nonlinear Anal.-Model Control, 23:6 (2018), 866–888  crossref  mathscinet  isi
    9. V. B. Bekezhanova, O. N. Goncharova, “Problems of evaporative convection (review)”, Fluid Dyn., 53:1 (2018), S69–S102  crossref  mathscinet  isi  elib  scopus
    10. K. Kaulakyte, N. Kloviene, K. Pileckas, “Nonhomogeneous boundary value problem for the stationary Navier-Stokes equations in a domain with a cusp”, Z. Angew. Math. Phys., 70:1 (2019), 36  crossref  mathscinet  zmath  isi  scopus
    11. A. Eismontaite, K. Pileckas, “On singular solutions of the initial boundary value problem for the Stokes system in a power cusp domain”, Appl. Anal., 98:13 (2019), 2400–2422  crossref  isi
    12. E. S. Baranovskii, “Optimal boundary control of nonlinear-viscous fluid flows”, Sb. Math., 211:4 (2020), 505–520  mathnet  crossref  crossref  mathscinet  isi  elib
    13. Ilin K., Morgulis A., “on the Stability of the Couette-Taylor Flow Between Rotating Porous Cylinders With Radial Flow”, Eur. J. Mech. B-Fluids, 80 (2020), 174–186  crossref  isi
    14. 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. Partial Differ. Equ., 59:1 (2020), 32  crossref  isi
    15. O. N. Shablovskii, “Sfericheskoe techenie idealnoi zhidkosti v prostranstvenno-neodnorodnom silovom pole”, Vestn. Tomsk. gos. un-ta. Matem. i mekh., 2020, no. 64, 146–155  mathnet  crossref
    16. Nowakowski B., Stroehmer G., “In-Flow and Out-Flow Problem For the Stokes System”, J. Math. Fluid Mech., 22:4 (2020), 58  crossref  mathscinet  zmath  isi
    17. Baranovskii E.S., Domnich A.A., “Model of a Nonuniformly Heated Viscous Flow Through a Bounded Domain”, Differ. Equ., 56:3 (2020), 304–314  crossref  mathscinet  zmath  isi
    18. Kaulakyte K., Kloviene N., “On Nonhomogeneous Boundary Value Problem For the Stationary Navier-Stokes Equations in a Symmetric Cusp Domain”, Math. Model. Anal., 26:1 (2021), 55–71  crossref  mathscinet  isi
    19. V. V. Zharinov, “Uravneniya Nave–Stoksa, algebraicheskii podkhod”, TMF, 209:3 (2021), 397–413  mathnet  crossref
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