Remarks on the weak formulation of the Navier–Stokes equations on the 2D hyperbolic space

Chan, Chi Hin; Czubak, Magdalena

doi:10.1016/j.anihpc.2015.01.002

Chan, Chi Hin ; Czubak, Magdalena

Annales de l'I.H.P. Analyse non linéaire, Tome 33 (2016) no. 3, pp. 655-698.

Résumé
Abstract

Les solutions de Leray–Hopf de l'équation Navier–Stokes sont connues pour être uniques sur $R^{2}$ . Dans nos travaux précédents, nous avons montré que ces solutions ne sont pas uniques si on se place dans un cadre hyperbolique. Dans cet article, nous montrons comment formuler le problème de façon à retrouver l'unicité.

The Leray–Hopf solutions to the Navier–Stokes equation are known to be unique on $R^{2}$ . In our previous work, we showed the breakdown of uniqueness in a hyperbolic setting. In this article, we show how to formulate the problem in order so the uniqueness can be restored.

Zbl

DOI : 10.1016/j.anihpc.2015.01.002

Classification : 76D05, 76D03
Mots clés : Navier–Stokes, Leray–Hopf, Non-uniqueness, Uniqueness, Hyperbolic space, Harmonic forms

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Chan, Chi Hin; Czubak, Magdalena. Remarks on the weak formulation of the Navier–Stokes equations on the 2D hyperbolic space. Annales de l'I.H.P. Analyse non linéaire, Tome 33 (2016) no. 3, pp. 655-698. doi : 10.1016/j.anihpc.2015.01.002. http://archive.numdam.org/articles/10.1016/j.anihpc.2015.01.002/

Bibliographie
Cité par

[1] Anderson, Michael T. The Dirichlet problem at infinity for manifolds of negative curvature, J. Differ. Geom., Volume 18 (1983) no. 4, pp. 701–721 | Zbl

[2] Anderson, Michael T.; Schoen, Richard Positive harmonic functions on complete manifolds of negative curvature, Ann. Math. (2), Volume 121 (1985) no. 3, pp. 429–461 | Zbl

[3] Avez, A.; Bamberger, Y. Mouvements sphériques des fluides visqueux incompressibles, J. Méc., Volume 17 (1978) no. 1, pp. 107–145 | Zbl

[4] Chan, Chi Hin; Czubak, Magdalena Non-uniqueness of the Leray–Hopf solutions in the hyperbolic setting, Dyn. Partial Differ. Equ., Volume 10 (2013) no. 1, pp. 43–77 | Zbl

[5] de Rham, Georges Differentiable manifolds, Forms, Currents, Harmonic Forms, Grundlehren Math. Wiss., Fundamental Principles of Mathematical Sciences, vol. 266, Springer-Verlag, Berlin, 1984 (Translated from French by F.R. Smith, With an introduction by S.S. Chern) | DOI | Zbl

[6] Dindoš, Martin; Mitrea, Marius The stationary Navier–Stokes system in nonsmooth manifolds: the Poisson problem in Lipschitz and $C^{1}$ domains, Arch. Ration. Mech. Anal., Volume 174 (2004) no. 1, pp. 1–47 | Zbl

[7] Dodziuk, Jozef $L^{2}$ harmonic forms on rotationally symmetric Riemannian manifolds, Proc. Am. Math. Soc., Volume 77 (1979) no. 3, pp. 395–400 | Zbl

[8] Ebin, David G.; Marsden, Jerrold Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. Math. (2), Volume 92 (1970), pp. 102–163 | Zbl

[9] Folland, Gerald B. Real Analysis. Modern Techniques and Their Applications, Pure Appl. Math. (New York), John Wiley & Sons Inc., New York, 1999 (A Wiley–Interscience Publication) | Zbl

[10] Hebey, Emmanuel Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities, Courant Lect. Notes Math., vol. 5, New York University Courant Institute of Mathematical Sciences, New York, 1999 | Zbl

[11] Heywood, John G. On uniqueness questions in the theory of viscous flow, Acta Math., Volume 136 (1976) no. 1–2, pp. 61–102 | Zbl

[12] Hopf, Eberhard Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen, Math. Nachr., Volume 4 (1951), pp. 213–231 | Zbl

[13] Il'in, A.A. Navier–Stokes and Euler equations on two-dimensional closed manifolds, Mat. Sb., Volume 181 (1990) no. 4, pp. 521–539 | Zbl

[14] Il'in, A.A.; Filatov, A.N. Unique solvability of the Navier–Stokes equations on a two-dimensional sphere, Dokl. Akad. Nauk SSSR, Volume 301 (1988) no. 1, pp. 18–22 | Zbl

[15] Jost, Jürgen Riemannian Geometry and Geometric Analysis, Universitext, Springer-Verlag, Berlin, 2008 | Zbl

[16] Khesin, Boris; Misiołek, Gerard The Euler and Navier–Stokes equations on the hyperbolic plane, Proc. Natl. Acad. Sci., Volume 109 (2012) no. 45, pp. 18324–18326

[17] Kodaira, Kunihiko Harmonic fields in Riemannian manifolds (generalized potential theory), Ann. Math. (2), Volume 50 (1949), pp. 587–665 | Zbl

[18] Ladyzhenskaya, O.A. The Mathematical Theory of Viscous Incompressible Flow, Math. Appl., vol. 2, Gordon and Breach Science Publishers, New York, 1969 (Translated from Russian by Richard A. Silverman and John Chu)

[19] Lee, John M. Riemannian Manifolds. An Introduction to Curvature, Grad. Texts Math., vol. 176, Springer-Verlag, New York, 1997 | Zbl

[20] Leray, Jean Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., Volume 63 (1934) no. 1, pp. 193–248 | JFM

[21] Mazzucato, Anna L. Besov–Morrey spaces: function space theory and applications to non-linear PDE, Trans. Am. Math. Soc., Volume 355 (2003) no. 4, pp. 1297–1364 | Zbl

[22] Mitrea, Marius; Taylor, Michael Navier–Stokes equations on Lipschitz domains in Riemannian manifolds, Math. Ann., Volume 321 (2001) no. 4, pp. 955–987 | Zbl

[23] Priebe, Volker Solvability of the Navier–Stokes equations on manifolds with boundary, Manuscr. Math., Volume 83 (1994) no. 2, pp. 145–159 | Zbl

[24] Sullivan, Dennis The Dirichlet problem at infinity for a negatively curved manifold, J. Differ. Geom., Volume 18 (1983) no. 4, pp. 723–732 | Zbl

[25] Taylor, Michael The Dirichlet problem on the hyperbolic ball http://www.unc.edu/math/Faculty/met/dphb.pdf (Preprint)

[26] Taylor, Michael E. Partial Differential Equations III. Nonlinear Equations, Appl. Math. Sci., vol. 117, Springer, New York, 2011 | Zbl

[27] Temam, Roger Navier–Stokes Equations. Theory and Numerical Analysis, Stud. Math. Appl., vol. 2, North-Holland Publishing Co., Amsterdam, 1984 (With an appendix by F. Thomasset) | Zbl

[28] Temam, Roger; Wang, Shou Hong Inertial forms of Navier–Stokes equations on the sphere, J. Funct. Anal., Volume 117 (1993) no. 1, pp. 215–242 | Zbl

[29] Yau, Shing Tung Harmonic functions on complete Riemannian manifolds, Commun. Pure Appl. Math., Volume 28 (1975), pp. 201–228 | Zbl

[30] Zhang, Qi S. The ill-posed Navier–Stokes equation on connected sums of $R^{3}$ , Complex Var. Elliptic Equ., Volume 51 (2006) no. 8–11, pp. 1059–1063 | Zbl

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