Infinite energy solutions of the two-dimensional Navier–Stokes equations

Thierry Gallay

doi:10.5802/afst.1558

Thierry Gallay ¹

¹ Univ. Grenoble Alpes, CNRS, Institut Fourier, 100 rue des Maths, 38610 Gières, France

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 26 (2017) no. 4, pp. 979-1027.

Résumé
Abstract

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 $ℝ^{2}$ 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].

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 $ℝ^{2}$ 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 $L^{\infty}$ 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].

Publié le : 2017-12-12

DOI : 10.5802/afst.1558

Affiliations des auteurs :

Thierry Gallay ¹

¹ Univ. Grenoble Alpes, CNRS, Institut Fourier, 100 rue des Maths, 38610 Gières, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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Thierry Gallay. Infinite energy solutions of the two-dimensional Navier–Stokes equations. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 26 (2017) no. 4, pp. 979-1027. doi : 10.5802/afst.1558. https://afst.centre-mersenne.org/articles/10.5802/afst.1558/

Bibliographie
Cité par

[1] Andrei L. Afendikov; Alexander Mielke Dynamical properties of spatially non-decaying 2D Navier-Stokes flows with Kolmogorov forcing in an infinite strip, J. Math. Fluid Mech., Volume 7 (2005), p. S51-S67 | DOI | Zbl

[2] David M. Ambrose; James P. Kelliher; Milton C. Lopes Filho; Helena J. Nussenzveig Lopes Serfati solutions to the 2D Euler equations on exterior domains, J. Differ. Equations, Volume 259 (2015) no. 9, pp. 4509-4560 | DOI | Zbl

[3] Jose M. Arrieta; Anibal Rodriguez-Bernal; Jan W. Cholewa; Tomasz Dlotko Linear parabolic equations in locally uniform spaces, Math. Models Methods Appl. Sci., Volume 14 (2004) no. 2, pp. 253-293 | DOI | Zbl

[4] Vladimir Chepyzhov; Sergey Zelik Infinite-energy solutions for dissipative Euler equations in $ℝ^{2}$ , J. Math. Fluid Mech., Volume 17 (2015) no. 3, pp. 513-532 | DOI | Zbl

[5] P. Collet A global existence result for the Navier-Stokes equation in the plane (1994) (unpublished manuscript)

[6] Peter Constantin; Ciprian Foias Navier-Stokes equations, Chicago Lectures in Mathematics, University of Chicago Press, 1988, ix+190 pages | Zbl

[7] Elaine Cozzi Vanishing viscosity in the plane for nondecaying velocity and vorticity, SIAM J. Math. Anal., Volume 41 (2009) no. 2, pp. 495-510 | DOI | Zbl

[8] Messoud A. Efendiev; Sergey Zelik The attractor for a nonlinear reaction-diffusion system in an unbounded domain, Commun. Pure Appl. Math., Volume 54 (2001) no. 6, pp. 625-688 | DOI | Zbl

[9] Klaus-Jochen Engel; Rainer Nagel One-parameter semigroups for linear evolution equations, Graduate Texts in Mathematics, 194, Springer, 2000, xxi+586 pages | Zbl

[10] Lawrence C. Evans Partial differential equations, Graduate Studies in Mathematics, 19, American Mathematical Society, 1998, xvii+662 pages | Zbl

[11] Eduard Feireisl Bounded, locally compact global attractors for semilinear damped wave equations on $ℝ^{n}$ , Differ. Integral Equ., Volume 9 (1996) no. 5, pp. 1147-1156 | Zbl

[12] Hiroshi P. Fujita; Tosio Kato On the Navier-Stokes initial value problem. I., Arch. Ration. Mech. Anal., Volume 16 (1964), pp. 269-315 | DOI | Zbl

[13] Giovanni P. Galdi An introduction to the mathematical theory of the Navier-Stokes equations. Steady-state problems, Springer Monographs in Mathematics, Springer, 2011, xiv+1018 pages | Zbl

[14] Thierry Gallay; Siniša Slijepčević Energy flow in formally gradient partial differential equations on unbounded domains, J. Dyn. Differ. Equations, Volume 13 (2001) no. 4, pp. 757-789 | DOI | Zbl

[15] Thierry Gallay; Siniša Slijepčević Energy bounds for the two-dimensional Navier-Stokes equations in an infinite cylinder, Commun. Partial Differ. Equations, Volume 39 (2014) no. 9, pp. 1741-1769 | DOI | Zbl

[16] Thierry Gallay; Siniša Slijepčević Uniform boundedness and long-time asymptotics for the two-dimensional Navier-Stokes equations in an infinite cylinder, J. Math. Fluid Mech., Volume 17 (2015) no. 1, pp. 23-46 | DOI | Zbl

[17] Thierry Gallay; C. Eugene Wayne Global stability of vortex solutions of the two-dimensional Navier-Stokes equation, Commun. Math. Phys., Volume 255 (2005) no. 1, pp. 97-129 | DOI | Zbl

[18] Yoshikazu Giga; Katsuya Inui; Shinʼya Matsui On the Cauchy problem for the Navier-Stokes equations with nondecaying initial data, Advances in fluid dynamics (Quaderni di Matematica.), Volume 4, Aracne, 1999, pp. 27-68 | Zbl

[19] Yoshikazu Giga; Shin?ya Matsui; Okihiro Sawada Global existence of two dimensional Navier-Stokes flow with non-decaying initial velocity, J. Math. Fluid Mech., Volume 3 (2001) no. 3, pp. 302-315 | DOI | Zbl

[20] Jean Ginibre; Giorgio Velo The Cauchy problem in local spaces for the complex Ginzburg-Landau equation, I: Compactness methods, Physica D, Volume 95 (1996) no. 2-4, pp. 191-228 | DOI | Zbl

[21] Dan Henry Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, 840, Springer, 1981, iv+348 pages | Zbl

[22] Tosio Kato The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Ration. Mech. Anal., Volume 58 (1975), pp. 181-205 | DOI | Zbl

[23] James P. Kelliher A characterization at infinity of bounded vorticity, bounded velocity solutions to the 2D Euler equations, Indiana Univ. Math. J., Volume 64 (2015) no. 6, pp. 1643-1666 | DOI | Zbl

[24] Hideo Kozono; Takayoshi Ogawa Two-dimensional Navier-Stokes flow in unbounded domains, Math. Ann., Volume 297 (1993) no. 1, pp. 1-31 | DOI | Zbl

[25] Olga Aleksandrovna Ladyzhenskaya Solution “in the large” of the nonstationary boundary value problem for the Navier-Stokes system with two space variables, Commun. Pure Appl. Math., Volume 12 (1959), pp. 427-433 | DOI | Zbl

[26] Jean Leray Étude de diverses équations intégrales non linéaires et de quelques problèmes que pose l’hydrodynamique, J. Math. Pures Appl., Volume 12 (1933), pp. 1-82 | Zbl

[27] Jean Leray Essai sur les mouvements plans d’un fluide visqueux que limitent des parois, J. Math. Pures Appl., Volume 13 (1934), pp. 331-418 | Zbl

[28] Elliott H. Lieb; Michael Loss Analysis, Graduate Studies in Mathematics, 14, American Mathematical Society, 1996, 278 pages | Zbl

[29] Pierre-Louis Lions Mathematical topics in fluid mechanics. Vol. 1. Incompressible models, Oxford Lecture Series in Mathematics and its Applications, 3, Clarendon Press, 1996, xiv+237 pages | Zbl

[30] Yasunori Maekawa; Yutaka Terasawa The Navier-Stokes equations with initial data in uniformly local $L^{p}$ spaces, Differ. Integral Equ., Volume 19 (2006) no. 4, pp. 369-400 | Zbl

[31] Andrew J. Majda; Andrea L. Bertozzi Vorticity and incompressible flow, Cambridge Texts in Applied Mathematics, Cambridge University Press, 2002, xii+545 pages | Zbl

[32] Kyûya Masuda Weak solutions of Navier-Stokes equations, Tohoku Math. J. (2), Volume 36 (1984), pp. 623-646 | DOI | Zbl

[33] Yves Meyer Ondelettes et opérateurs. II. Opérateurs de Calderón-Zygmund, Actualités Mathématiques, Hermann, 1990 | Zbl

[34] Alexander Mielke; Guido Schneider Attractors for modulation equations on unbounded domains — existence and comparison, Nonlinearity, Volume 8 (1995) no. 5, pp. 743-768 | DOI | Zbl

[35] Ammon Pazy Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, 44, Springer, 1983, viii+279 pages | Zbl

[36] Murray H. Protter; Hans F. Weinberger Maximum principles in differential equations, Prentice-Hall Partial Differential Equations Series, Prentice-Hall, 1967, x+261 pages | Zbl

[37] Michael Reed; Barry Simon Methods of modern mathematical physics. I. Functional analysis, Academic Press, 1972, xvii+325 pages | Zbl

[38] Michael Reed; Barry Simon Methods of modern mathematical physics. II. Fourier analysis, self-adjointness, Academic Press, 1975, xv+361 pages | Zbl

[39] Okihiro Sawada; Yasushi Taniuchi A remark on $L^{\infty}$ solutions to the 2-D Navier-Stokes equations, J. Math. Fluid Mech., Volume 9 (2007) no. 4, pp. 533-542 | DOI | Zbl

[40] Maria E. Schonbek Large time behaviour of solutions to the Navier-Stokes equations, Commun. Partial Differ. Equations, Volume 11 (1986), pp. 733-763 | DOI | Zbl

[41] Philippe Serfati Solutions $C^{\infty}$ en temps, $n$ -log Lipschitz bornées en espace et équation d’Euler, C. R. Acad. Sci., Paris, Volume 320 (1995) no. 5, pp. 555-558 | Zbl

[42] Elias M. Stein Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, 43, Princeton University Press, 1993, xiii+695 pages | Zbl

[43] Roger Temam Navier-Stokes equations. Theory and numerical analysis., Studies in Mathematics and its Applications, 2, North-Holland, 1984, xii+526 pages | Zbl

[44] Michael Wiegner Decay results for weak solutions of the Navier-Stokes equations on $ℝ^{n}$ , J. Lond. Math. Soc., Volume 35 (1987), pp. 303-313 | DOI | Zbl

[45] Sergey Zelik Infinite energy solutions for damped Navier-Stokes equations in $ℝ^{2}$ , J. Math. Fluid Mech., Volume 15 (2013) no. 4, pp. 717-745 | DOI | Zbl

Dimitri Cobb Bounded solutions in incompressible hydrodynamics, Journal of Functional Analysis, Volume 286 (2024) no. 5, p. 110290 | DOI:10.1016/j.jfa.2023.110290
David M. Ambrose; Fazel Hadadifard; James P. Kelliher Contour Dynamics and Global Regularity for Periodic Vortex Patches and Layers, SIAM Journal on Mathematical Analysis, Volume 56 (2024) no. 2, p. 2286 | DOI:10.1137/22m1525818
Zachary Bradshaw; Tai-Peng Tsai On the Local Pressure Expansion for the Navier–Stokes Equations, Journal of Mathematical Fluid Mechanics, Volume 24 (2022) no. 1 | DOI:10.1007/s00021-021-00637-4
Ken Abe On the large time L∞-estimates of the Stokes semigroup in two-dimensional exterior domains, Journal of Differential Equations, Volume 300 (2021), p. 337 | DOI:10.1016/j.jde.2021.08.007
Ken Abe Vanishing viscosity limits for axisymmetric flows with boundary, Journal de Mathématiques Pures et Appliquées, Volume 137 (2020), p. 1 | DOI:10.1016/j.matpur.2020.01.005

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