With cedram.org All online volumes Latest online issues Advanced Search Table of contents for this issue | Previous article | Next article Thierry Gallay; Luis Miguel RodriguesSur le temps de vie de la turbulence bidimensionnelleAnnales de la faculté des sciences de Toulouse Sér. 6, 17 no. 4 (2008), p. 719-733, doi: 10.5802/afst.1199 Article PDF | Reviews MR 2499852 | Zbl 1159.76019 Résumé - AbstractIt is known that all solutions of the two-dimensional Navier-Stokes equation whose vorticity distribution is integrable converge as $t\rightarrow \infty$ to a self-similar flow called Oseen’s vortex. In this article we estimate the time needed for a solution to reach a neighborhood of Oseen’s vortex, starting from arbitrary but well localized initial data. We thus obtain an upper bound on the lifetime of the two-dimensional freely decaying turbulence, depending on the initial Reynolds number. The particular cases where the vorticity distribution is either nonnegative, or sufficiently close to Oseen’s vortex, are studied in more detail. Bibliography[1] Arnold (A.), Markowich (P.), Toscani (G.) et Unterreiter (A.).— On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations, Comm. Partial Differential Equations 26, p. 43-100 (2001).  MR 1842428 |  Zbl 0982.35113[2] Ben-Artzi (B.).— Global solutions of two-dimensional Navier-Stokes and Euler equations, Arch. Rational Mech. Anal. 128, p. 329-358 (1994).  MR 1308857 |  Zbl 0837.35110[3] Brezis (H.).— Remarks on the preceding paper by M. Ben-Artzi : « Global solutions of two-dimensional Navier-Stokes and Euler equations », Arch. Rational Mech. Anal. 128, p. 359-360 (1994).  MR 1308858 |  Zbl 0837.35112[4] Carlen (E. A.) et Loss (L.).— Optimal smoothing and decay estimates for viscously damped conservation laws, with applications to the $2$-D Navier-Stokes equation, Duke Math. J. 81, p. 135-157 (1996). Article |  MR 1381974 |  Zbl 0859.35011[5] Gallay (Th.) et Wayne (C. E.).— Invariant manifolds and the long-time asymptotics of the Navier-Stokes and vorticity equations on ${\bf R}^2$, Arch. Ration. Mech. Anal. 163, p. 209-258 (2002).  MR 1912106 |  Zbl 1042.37058[6] Gallay (Th.) et Wayne (C. E.).— Global stability of vortex solutions of the two-dimensional Navier-Stokes equation, Comm. Math. Phys. 255, p. 97-129 (2005).  MR 2123378 |  Zbl 1139.35084[7] Gallay (Th.) et Wayne (C. E.).— Existence and Stability of Asymmetric Burgers Vortices, J. Math. Fluid Mechanics 9, p. 243-261 (2007).  MR 2329268 |  Zbl 1119.76012 Search for an article Search within the site