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Leray’s Problem on Existence of Steady State Solutions for the Navier-Stokes Flow

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Handbook of Mathematical Analysis in Mechanics of Viscous Fluids

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Abstract

This is a survey of results on the Leray problem (1933) for the nonhomogeneous boundary value problem for the steady Navier–Stokes equations in a bounded domain with multiple boundary components. The boundary conditions are assumed only to satisfy the necessary requirement of zero total flux. The authors have proved that the problem is solvable in arbitrary bounded planar or three-dimensional 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. Similar existence results (without any restrictions on fluxes) are proved for steady Navier–Stokes system in two- and three-dimensional exterior domains with multiply connected boundary under assumptions of axial symmetry. In particular, it was shown that in domains with two axes of symmetry and for symmetric boundary datum, the two-dimensional exterior problem has a symmetric solution vanishing at infinity.

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Acknowledgements

The research of K. Pileckas leading to these results has received funding from the Lithuanian-Swiss Cooperation Programme to reduce economic and social disparities within the enlarged European Union under project agreement No. CH-3-SMM-01/01.

The research of M. V. Korobkov was partially supported by the Russian Foundation for Basic Research (Grant No. 14-01-00768-a), by the Grant of the Russian Federation for the State Support of Researches (Agreement No. 14.B25.31.0029), and by the Dynasty Foundation.

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Authors and Affiliations

  1. Sobolev Institute of Mathematics, Acad. Koptyuga pr., 4, 630090, Novosibirsk, Russia

    Mikhail V. Korobkov

  2. Novosibirsk State University, Pirogova Str. 2, 630090, Novosibirsk, Russia

    Mikhail V. Korobkov

  3. Faculty of Mathematics and Informatics, Vilnius University, Naugarduko Str., 24, 03225, Vilnius, Lithuania

    Konstantin Pileckas

  4. Department of Mathematics and Physics, Second University of Naples, Lincoln Avenue, 5 - 81100, Caserta, Italy

    Remigio Russo

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  1. Mikhail V. Korobkov
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  2. Konstantin Pileckas
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  3. Remigio Russo
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Correspondence to Mikhail V. Korobkov .

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Editors and Affiliations

  1. Tokyo, Tokyo, Japan

    Yoshikazu Giga

  2. Toulon, Var, France

    Antonin Novotny

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Korobkov, M.V., Pileckas, K., Russo, R. (2016). Leray’s Problem on Existence of Steady State Solutions for the Navier-Stokes Flow. In: Giga, Y., Novotny, A. (eds) Handbook of Mathematical Analysis in Mechanics of Viscous Fluids. Springer, Cham. https://doi.org/10.1007/978-3-319-10151-4_5-1

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