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Russian Mathematical Surveys, 2014, Volume 69, Issue 6, Pages 1065–1122
DOI: https://doi.org/10.1070/RM2014v069n06ABEH004928 (Mi rm9616) 		 
This article is cited in 39 scientific papers (total in 39 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
English version PDF (1076 kB) Citations (39)
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DOI: https://doi.org/10.1070/RM2014v069n06ABEH004928
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).
Received: 20.08.2014
Russian version:
Uspekhi Matematicheskikh Nauk, 2014, Volume 69, Issue 6(420), Pages 115–176
DOI: https://doi.org/10.4213/rm9616
Bibliographic databases:
Document Type: Article
UDC: 517.59
MSC: Primary 35Q30, 35Q31, 76D05; Secondary 76D07, 76D10
Language: English
Original paper language: Russian
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
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