Some Liouville theorems for stationary Navier-Stokes equations in Lebesgue and Morrey spaces

Chamorro, Diego; Jarrín, Oscar; Lemarié-Rieusset, Pierre-Gilles

doi:10.1016/j.anihpc.2020.08.006

Chamorro, Diego ¹ ; Jarrín, Oscar ² ; Lemarié-Rieusset, Pierre-Gilles ¹

¹ a Laboratoire de Mathématiques et Modélisation d'Evry (LaMME) - UMR 8071, Université d'Evry Val d'Essonne, 23 Boulevard de France, 91037 Evry Cedex, France
² b Dirección de investigación y desarrollo (DIDE), Universidad Técnica de Ambato, Avenida de los Chasquis, 180207, Ambato, Ecuador

Annales de l'I.H.P. Analyse non linéaire, mai – juin 2021, Tome 38 (2021) no. 3, pp. 689-710.

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Résumé

Uniqueness of Leray solutions of the 3D Navier-Stokes equations is a challenging open problem. In this article we will study this problem for the 3D stationary Navier-Stokes equations in the whole space $R^{3}$ . Under some additional hypotheses, stated in terms of Lebesgue and Morrey spaces, we will show that the trivial solution $\vec{U} = 0$ is the unique solution. This type of results are known as Liouville theorems.

MR Zbl

DOI : 10.1016/j.anihpc.2020.08.006

Mots-clés : Navier–Stokes equations, Stationary system, Liouville theorem, Morrey spaces

Affiliations des auteurs :

Chamorro, Diego ¹ ; Jarrín, Oscar ² ; Lemarié-Rieusset, Pierre-Gilles ¹

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     author = {Chamorro, Diego and Jarr{\'\i}n, Oscar and Lemari\'e-Rieusset, Pierre-Gilles},
     title = {Some {Liouville} theorems for stationary {Navier-Stokes} equations in {Lebesgue} and {Morrey} spaces},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {689--710},
     publisher = {Elsevier},
     volume = {38},
     number = {3},
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     url = {https://www.numdam.org/articles/10.1016/j.anihpc.2020.08.006/}
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Chamorro, Diego; Jarrín, Oscar; Lemarié-Rieusset, Pierre-Gilles. Some Liouville theorems for stationary Navier-Stokes equations in Lebesgue and Morrey spaces. Annales de l'I.H.P. Analyse non linéaire, mai – juin 2021, Tome 38 (2021) no. 3, pp. 689-710. doi : 10.1016/j.anihpc.2020.08.006. https://www.numdam.org/articles/10.1016/j.anihpc.2020.08.006/

Bibliographie
Cité par

[1] Chae, D.; Yoneda, T. On the Liouville theorem for the stationary Navier-Stokes equations in a critical space, J. Math. Anal. Appl., Volume 405 (2013), pp. 706-710 | DOI | MR | Zbl

[2] Chae, D.; Wolf, J. On Liouville type theorems for the steady Navier- Stokes equations in $R^{3}$ , 2016 | arXiv | MR | Zbl

[3] Chae, G.; Weng, S. Liouville type theorems for the steady axially symmetric Navier-Stokes and magnetohydrodynamic equations, Discrete Contin. Dyn. Syst., Volume 36 (2016) no. 10, pp. 5267-5285 | DOI | MR | Zbl

[4] Galdi, G.P. An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Steady-State Problems, Springer Monographs in Mathematics, Springer, New York, 2011 | MR | Zbl

[5] Gérard, P.; Meyer, Y.; Oru, F. Inégalités de Sobolev précisées, Sémin. Équ. Dériv. Partielles, Volume 1996–1997 ( 1996–1997 ), pp. 1-8 | Numdam | MR | Zbl

[6] Jarrín, O. A remark on the Liouville problem for stationary Navier-Stokes equations in Lorentz and Morrey spaces, J. Math. Anal. Appl., Volume 486 (2020) no. 1 | DOI | MR | Zbl

[7] Jarrín, O. A short note on the uniqueness of the trivial solution for the steady-state Navier-Stokes equations, 2019 | arXiv

[8] Koch, G.; Nadirashvili, N.; Seregin, G.; Sverak, V. Liouville theorems for the Navier-Stokes equations and applications, Acta Math., Volume 203 (2009), pp. 83-105 | DOI | MR | Zbl

[9] Kozono, H.; Terasawa, Y.; Wakasugi, Y. A remark on Liouville-type theorems for the stationary Navier-Stokes equations in three space dimensions, J. Funct. Anal., Volume 272 (2017), pp. 804-818 | DOI | MR | Zbl

[10] Lemarié-Rieusset, P.G. The Navier-Stokes Problem in the 21st Century, Chapman & Hall/CRC, 2016 | DOI | MR | Zbl

[11] Lemarié-Rieusset, P.G. Recent Developments in the Navier-Stokes Problem, Chapman & Hall/CRC, 2002 | MR | Zbl

[12] Seregin, G. Liouville type theorem for stationary Navier-Stokes equations, Nonlinearity, Volume 29 (2015) (2191 :2195) | MR

[13] Seregin, G. A Liouville type theorem for steady-state Navier-Stokes equations, 2016 | arXiv | MR | Zbl

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