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Infinite energy solutions of the two-dimensional Navier–Stokes equations
Thierry Gallay
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 26 (2017) no. 4, p. 979-1027

These notes are based on a series of lectures delivered by the author at the University of Toulouse in February 2014. They are entirely devoted to the initial value problem and the long-time behavior of solutions for the two-dimensional incompressible Navier–Stokes equations, in the particular case where the domain occupied by the fluid is the whole plane  2 and the velocity field is only assumed to be bounded. In this context, local well-posedness is not difficult to establish [18], and a priori estimates on the vorticity distribution imply that all solutions are global and grow at most exponentially in time [19, 39]. Moreover, as was recently shown by S. Zelik, localized energy estimates can be used to obtain a much better control on the uniformly local energy norm of the velocity field [45]. The aim of these notes is to present, in an explanatory and self-contained way, a simplified and optimized version of Zelik’s argument which, in combination with a new formulation of the Biot–Savart law for bounded vorticities, allows one to show that the L norm of the velocity field grows at most linearly in time. The results do not rely on the viscous dissipation, and remain therefore valid for the so-called “Serfati solutions” of the two-dimensional Euler equations [2]. In the viscous case, a recent work by S. Slijepčević and the author shows that all solutions stay uniformly bounded if the velocity field and the pressure are periodic in a given space direction [15, 16].

Ces notes s’appuient sur un cours donné par l’auteur à l’université de Toulouse en février 2014. Elles sont entièrement consacrées au problème de Cauchy et au comportement en temps grands des solutions des équations de Navier–Stokes incompressibles à deux dimensions, dans le cas particulier où le domaine occupé par le fluide est le plan 2 tout entier et où le champ de vitesse est seulement supposé borné. Dans ce contexte, il n’est pas difficile de montrer que le problème est localement bien posé [18], et des estimations a priori sur le tourbillon impliquent que toutes les solutions sont globales et que leur croissance temporelle est au plus exponentielle [19, 39]. En outre, comme l’a récemment montré S. Zelik, on peut utiliser des estimations d’énergie localisées pour obtenir un contrôle beaucoup plus précis sur le champ de vitesse dans l’espace d’énergie uniformément local [45]. Le but de ces notes est de présenter, de façon pédagogique et indépendante, une version simplifiée et optimisée de l’argument de Zelik qui, combinée avec une nouvelle formulation de la loi de Biot–Savart pour des tourbillons bornés, permet de montrer que la croissance temporelle du champ de vitesse est au plus linéaire. Ces résultats sont établis sans utiliser la dissipation d’énergie due à la viscosité, et restent donc valables pour les solutions dites « de Serfati » des équations d’Euler en dimension deux [2]. Dans le cas visqueux, un travail récent de S. Slijepčević et de l’auteur montre que toutes les solutions demeurent uniformément bornées si le champ de vitesse et la pression sont périodiques dans une direction donnée du plan [15, 16].

Published online : 2017-12-13
DOI : https://doi.org/10.5802/afst.1558
@article{AFST_2017_6_26_4_979_0,
     author = {Thierry Gallay},
     title = {Infinite energy solutions of the two-dimensional Navier--Stokes equations},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 26},
     number = {4},
     year = {2017},
     pages = {979-1027},
     doi = {10.5802/afst.1558},
     language = {en},
     url = {https://afst.centre-mersenne.org/item/AFST_2017_6_26_4_979_0}
}
Gallay, Thierry. Infinite energy solutions of the two-dimensional Navier–Stokes equations. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 26 (2017) no. 4, pp. 979-1027. doi : 10.5802/afst.1558. afst.centre-mersenne.org/item/AFST_2017_6_26_4_979_0/

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