Leray's problem for the Navier–Stokes equations revisited

Joyce C. Rigelo; Lineia Schütz; Janaína P. Zingano; Paulo R. Zingano

doi:10.1016/j.crma.2016.02.008

Partial differential equations

Leray's problem for the Navier–Stokes equations revisited

Presented by: Haïm Brézis

Joyce C. Rigelo ¹; Lineia Schütz ²; Janaína P. Zingano ²; Paulo R. Zingano ²

¹ Department of Petroleum and Geosystems Engineering, The University of Texas at Austin, Austin, TX 78712, USA
² Departamento de Matemática Pura e Aplicada, Universidade Federal do Rio Grande do Sul, Porto Alegre, RS 91509, Brazil

Comptes Rendus. Mathématique, Volume 354 (2016) no. 5, pp. 503-509.

Abstract
Résumé

In a seminal work in 1934, J. Leray constructed solutions $u (\cdot, t) \in L^{\infty} ([0, \infty), L_{σ}^{2} (R^{3})) \cap C_{w}^{0} ([0, \infty), L^{2} (R^{3})) \cap L^{2} ([0, \infty), {\dot{H}}^{1} (R^{3}))$ of the Navier–Stokes equations for arbitrary initial data $u (\cdot, 0) \in L_{σ}^{2} (R^{3})$ and left it open whether ${‖ u (\cdot, t) ‖}_{L^{2} (R^{3})}$ would necessarily tend to zero as $t \to \infty$ . This question was answered positively fifty years later by T. Kato, using a different approach. Here, we reexamine Leray's problem and solve this and other important related questions using Leray's original ideas and some standard tools (Fourier transform, Duhamel's principle, heat kernel estimates) already in use in his time.

En 1934, J. Leray a construit des solutions faibles $u (\cdot, t) \in L^{\infty} ([0, \infty), L_{σ}^{2} (R^{3})) \cap C_{w}^{0} ([0, \infty), L^{2} (R^{3})) \cap L^{2} ([0, \infty), {\dot{H}}^{1} (R^{3}))$ pour les équations de Navier–Stokes avec des données initiales $u (\cdot, 0) \in L_{σ}^{2} (R^{3})$ arbitraires, où il a laissé non résolue la question de savoir si ${‖ u (\cdot, t) ‖}_{L^{2} (R^{3})}$ tendrait toujours vers zéro quand $t \to \infty$ , à laquelle a été répondu positivement en 1984 par T. Kato, au moyen d'une autre approche. Ici, on reconsidère le problème de Leray et quelques-unes de ses extensions, qui sont résolus en n'employant que des idées développées par Leray en 1934 et des techniques classiques, très utilisées déjà à cette époque.

Received: 2015-07-06
Accepted: 2016-02-23
Published online: 2016-03-29

DOI: 10.1016/j.crma.2016.02.008

Author's affiliations:

Joyce C. Rigelo ¹; Lineia Schütz ²; Janaína P. Zingano ²; Paulo R. Zingano ²

¹ Department of Petroleum and Geosystems Engineering, The University of Texas at Austin, Austin, TX 78712, USA
² Departamento de Matemática Pura e Aplicada, Universidade Federal do Rio Grande do Sul, Porto Alegre, RS 91509, Brazil

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     author = {Joyce C. Rigelo and Lineia Sch\"utz and Jana{\'\i}na P. Zingano and Paulo R. Zingano},
     title = {Leray's problem for the {Navier{\textendash}Stokes} equations revisited},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {503--509},
     publisher = {Elsevier},
     volume = {354},
     number = {5},
     year = {2016},
     doi = {10.1016/j.crma.2016.02.008},
     language = {en},
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Joyce C. Rigelo; Lineia Schütz; Janaína P. Zingano; Paulo R. Zingano. Leray's problem for the Navier–Stokes equations revisited. Comptes Rendus. Mathématique, Volume 354 (2016) no. 5, pp. 503-509. doi : 10.1016/j.crma.2016.02.008. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2016.02.008/

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