Remarks on Liouville type theorems for steady-state Navier–Stokes equations
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- by G. Seregin
- St. Petersburg Math. J. 30 (2019), 321-328
- DOI: https://doi.org/10.1090/spmj/1544
- Published electronically: February 14, 2019
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Abstract:
Liouville type theorems for the stationary Navier–Stokes equations are proved under certain assumptions. These assumptions are motivated by conditions that appear in Liouville type theorems for the heat equations with a given divergence free drift.References
- J. P. Bourguignon and H. Brezis, Remarks on the Euler equation, J. Functional Analysis 15 (1974), 341–363. MR 0344713, DOI 10.1016/0022-1236(74)90027-5
- Dongho Chae, Liouville-type theorems for the forced Euler equations and the Navier-Stokes equations, Comm. Math. Phys. 326 (2014), no. 1, 37–48. MR 3162482, DOI 10.1007/s00220-013-1868-x
- Dongho Chae and Tsuyoshi Yoneda, On the Liouville theorem for the stationary Navier-Stokes equations in a critical space, J. Math. Anal. Appl. 405 (2013), no. 2, 706–710. MR 3061045, DOI 10.1016/j.jmaa.2013.04.040
- Dongho Chae and Jörg Wolf, On Liouville type theorems for the steady Navier-Stokes equations in $\Bbb {R}^3$, J. Differential Equations 261 (2016), no. 10, 5541–5560. MR 3548261, DOI 10.1016/j.jde.2016.08.014
- G. P. Galdi, An introduction to the mathematical theory of the Navier-Stokes equations, 2nd ed., Springer Monographs in Mathematics, Springer, New York, 2011. Steady-state problems. MR 2808162, DOI 10.1007/978-0-387-09620-9
- Mariano Giaquinta, Multiple integrals in the calculus of variations and nonlinear elliptic systems, Annals of Mathematics Studies, vol. 105, Princeton University Press, Princeton, NJ, 1983. MR 717034
- D. Gilbarg and H. F. Weinberger, Asymptotic properties of steady plane solutions of the Navier-Stokes equations with bounded Dirichlet integral, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 5 (1978), no. 2, 381–404. MR 501907
- Gabriel Koch, Nikolai Nadirashvili, Gregory A. Seregin, and Vladimir Šverák, Liouville theorems for the Navier-Stokes equations and applications, Acta Math. 203 (2009), no. 1, 83–105. MR 2545826, DOI 10.1007/s11511-009-0039-6
- A. I. Nazarov and N. N. Ural′tseva, The Harnack inequality and related properties of solutions of elliptic and parabolic equations with divergence-free lower-order coefficients, Algebra i Analiz 23 (2011), no. 1, 136–168 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 23 (2012), no. 1, 93–115. MR 2760150, DOI 10.1090/S1061-0022-2011-01188-4
- G. Seregin, Liouville type theorem for stationary Navier-Stokes equations, Nonlinearity 29 (2016), no. 8, 2191–2195. MR 3538409, DOI 10.1088/0951-7715/29/8/2191
- —, A Liouville type theorem for steady-state Navier–Stokes equations, arXiv:1611.01563.
- Gregory Seregin, Luis Silvestre, Vladimír Šverák, and Andrej Zlatoš, On divergence-free drifts, J. Differential Equations 252 (2012), no. 1, 505–540. MR 2852216, DOI 10.1016/j.jde.2011.08.039
- Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
Bibliographic Information
- G. Seregin
- Affiliation: University of Oxford, UK; St. Petersburg Department of the Steklov Mathematical Institute, St. Petersburg, Russia
- Email: seregin@maths.ox.ac.uk
- Received by editor(s): July 24, 2017
- Published electronically: February 14, 2019
- Additional Notes: Supported by RFBR (grant no. 17-01-00099a)
- © Copyright 2019 American Mathematical Society
- Journal: St. Petersburg Math. J. 30 (2019), 321-328
- MSC (2010): Primary 35Q30
- DOI: https://doi.org/10.1090/spmj/1544
- MathSciNet review: 3790738
Dedicated: Dedicated to the 130th anniversary of Vladimir Ivanovich Smirnov