zbMATH — the first resource for mathematics

Dynamic stability of the three-dimensional axisymmetric Navier-Stokes equations with swirl. (English) Zbl 1138.35077
Summary: We study the dynamic stability of the three-dimensional axisymmetric Navier-Stokes equations with swirl. To this purpose, we propose a new one-dimensional model that approximates the Navier-Stokes equations along the symmetry axis. An important property of this one-dimensional model is that one can construct from its solutions a family of exact solutions of the three-dimensional Navier-Stokes equations. The nonlinear structure of the one-dimensional model has some very interesting properties. On one hand, it can lead to tremendous dynamic growth of the solution within a short time. On the other hand, it has a surprising dynamic depletion mechanism that prevents the solution from blowing up in finite time. By exploiting this special nonlinear structure, we prove the global regularity of the three-dimensional Navier-Stokes equations for a family of initial data, whose solutions can lead to large dynamic growth, but yet have global smooth solutions.

35Q30 Navier-Stokes equations
35B35 Stability in context of PDEs
35B65 Smoothness and regularity of solutions to PDEs
35B45 A priori estimates in context of PDEs
Full Text: DOI
[1] Babin, Indiana Univ Math J 50 pp 1– (2001) · Zbl 1013.35065 · doi:10.1512/iumj.2001.50.2155
[2] Beale, Comm Math Phys 94 pp 61– (1984)
[3] Caffarelli, Comm Pure Appl Math 35 pp 771– (1982)
[4] Cao, Ann of Math (2) 165 pp 1– (2007)
[5] Chae, Adv Math 203 pp 497– (2006)
[6] Chae, Math Z 239 pp 645– (2002)
[7] Constantin, Commun Partial Differential Equations 21 pp 559– (1996) · Zbl 0853.35091 · doi:10.1080/03605309608821197
[8] ; Navier-Stokes equations. Chicago University Press, Chicago, 1988. · Zbl 0687.35071
[9] Constantin, Comm Pure Appl Math 38 pp 715– (1985)
[10] Deng, Comm Partial Differential Equations 30 pp 225– (2005)
[11] Deng, Comm Partial Differential Equations 31 pp 293– (2006)
[12] Existence and smoothness of the Navier-Stokes equation. The millennium prize problems, 57–67. Clay Mathematics Institute, Cambridge, Mass.. 2006.
[13] Available online at http://www.claymath.org/millenium/Navier-Stokes_Equations/
[14] Gibbon, Nonlinearity 19 pp 1969– (2006)
[15] Hou, Discrete Contin Dyn Syst 12 pp 1– (2005)
[16] Hou, J Nonlinear Sci 16 pp 639– (2006)
[17] Kerr, Phys Fluids 5 pp 1725– (1993)
[18] 2nd English ed., revised and enlarged. The mathematical theory of viscous incompressible flow. Mathematics and Its Applications, 2. Gordon and Breach, New York–London–Paris, 1969.
[19] Lin, Comm Pure Appl Math 51 pp 241– (1998)
[20] Liu, SIAM J Numer Anal 44 pp 2456–
[21] ; Vorticity and incompressible flow. Cambridge University Press, Cambridge, 2002. · Zbl 0983.76001
[22] Navier-Stokes equations. American Mathematical Society, Providence, R.I., 2001. · doi:10.1090/chel/343
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.