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Uspekhi Mat. Nauk, 2018, Volume 73, Issue 4(442), Pages 103–170 (Mi umn9822)  

This article is cited in 9 scientific papers (total in 9 papers)

Liouville-type theorems for the Navier–Stokes equations

G. A. Sereginab, T. N. Shilkinab

a St. Petersburg Department of the Steklov Mathematical Institute
b Voronezh State University

Abstract: An approach to the study of local regularity of weak solutions of the Navier–Stokes equations is described which is based on the reduction of questions of local smoothness of the original solutions to the proof of Liouville-type theorems for bounded ancient solutions of it. A survey is also given of results on Liouville theorems that are known at present for various classes of ancient solutions of the Navier–Stokes equations.
Bibliography: 55 titles.

Keywords: Navier–Stokes equations, ancient solutions, Liouville theorems.

Funding Agency Grant Number
Ministry of Education and Science of the Russian Federation 14.Z50.31.0037
This research was carried out with the financial support of the Ministry of Education and Science of the Russian Federation (project no. 14.Z50.31.0037).


DOI: https://doi.org/10.4213/rm9822

Full text: PDF file (1016 kB)
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English version:
Russian Mathematical Surveys, 2018, 73:4, 661–724

Bibliographic databases:

UDC: 517.958:531.32
MSC: Primary 35B53, 35Q30; Secondary 35D30
Received: 26.03.2018

Citation: G. A. Seregin, T. N. Shilkin, “Liouville-type theorems for the Navier–Stokes equations”, Uspekhi Mat. Nauk, 73:4(442) (2018), 103–170; Russian Math. Surveys, 73:4 (2018), 661–724

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Chernobai M., Shilkin T., “Scalar Elliptic Equations With a Singular Drift”, Complex Var. Elliptic Equ.  crossref  isi
    2. M. Chernobay, “On type I blow up for the Navier–Stokes equations near the boundary”, Kraevye zadachi matematicheskoi fiziki i smezhnye voprosy teorii funktsii. 47, K 85-letiyu Vsevoloda Alekseevicha SOLONNIKOVA, Zap. nauchn. sem. POMI, 477, POMI, SPb., 2018, 136–149  mathnet
    3. D. Albritton, T. Barker, “On local type I singularities of the Navier-Stokes equations and Liouville theorems”, J. Math. Fluid Mech., 21:3 (2019), 43, 11 pp.  crossref  mathscinet  isi  scopus
    4. A. V. Chernov, “On preservation of global solvability of controlled second kind operator equation”, Ufa Math. J., 12:1 (2020), 56–81  mathnet  crossref  isi
    5. A. V. Chernov, “O totalno globalnoi razreshimosti upravlyaemogo operatornogo uravneniya vtorogo roda”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 30:1 (2020), 92–111  mathnet  crossref
    6. A. T. Il'ichev, “Dynamics and spectral stability of soliton-like structures in fluid-filled membrane tubes”, Russian Math. Surveys, 75:5 (2020), 843–882  mathnet  crossref  crossref  mathscinet  isi  elib
    7. G. Seregin, “Local regularity of axisymmetric solutions to the Navier-Stokes equations”, Anal. Math. Phys., 10:4 (2020), 46  crossref  mathscinet  zmath  isi
    8. D. Albritton, T. Barker, “Localised necessary conditions for singularity formation in the Navier-Stokes equations with curved boundary”, J. Differ. Equ., 269:9 (2020), 7529–7573  crossref  mathscinet  zmath  isi
    9. A. V. Chernov, “O sokhranenii globalnoi razreshimosti operatornogo uravneniya pervogo roda s upravlyaemoi dobavochnoi nelineinostyu”, Materialy Voronezhskoi vesennei matematicheskoi shkoly Sovremennye metody teorii kraevykh zadach. Pontryaginskie chteniyaXXX. Voronezh, 39 maya 2019 g. Chast 3, Itogi nauki i tekhn. Ser. Sovrem. mat. i ee pril. Temat. obz., 192, VINITI RAN, M., 2021, 131–141  mathnet  crossref
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