Abstract
In this paper we study the Navier-Stokes flow on the two-dimensional torusS 1 ×S 1 excited by the external force (k 2 sinky, 0) and find the long-time behavior for the flow starting from some states, whereS 1=[0,2π](mod 2π). Especially for the casek=2, it follows from an analysis and computation that the Navier-Stokes flow with the initial state cos(mx+ny) or sin(mx+ny) will likely evolve through at most one step bifurcation to either a steady-state solution or a time-dependent periodic solution for any Reynolds number and integersm andn.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
V. L. Arnold and L. D. Meshalkin, A. N. Kolmogorov's seminar on selected problems of analysis (1958–1959),Russ. Math. Surv. 15(1):247–250 (1960).
A. V. Babin and M. L. Vishik, Attractors of partial differential evolution equations and estimates of their dimension,Russ. Math. Surv. 38:151–213 (1983).
Z. M. Chen and W. G. Price, Time dependent periodic Navier-Stokes flows on a two-dimensional torus,Commun. Math. Phys. 179:577–597 (1996).
P. Constantin, C. Foias, and R. Temam, On the large time Galerkin approximation of the Navier-Stokes equations,SIAM J. Num. Anal. 21:615–634 (1984).
V. Franceschini, C. Giberti, and M. Nicolini, Common periodic behavior in larger truncation of the Navier-Stokes equations,J. Stat. Phys. 50:879–896 (1988).
D. Gilbarg and N. S. Trudinger,Elliptic Partial Differential Equations of Second Order (Springer-Verlag, Berlin, 1977).
J. Guckenheimer and P. Holmes,Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer-Verlag, Berlin, 1983).
V. I. Iudovich, Example of the generation of a secondary stationary or periodic flow when there is loss of stability of the laminar flow of a viscous incompressible fluid,J. Math. Mech. 29(3):453–467 (1965).
M. S. Jolly, Bifurcation computations on an approximate inertial manifold for the 2-D Navier-Stokes equations,Physica D 63:8–20 (1993).
A. Krasnosel'skii,Topological Methods in the Theory of Non-Linear Integral Equations (Pergamon, New York, 1964).
J. D. Lambert,Numerical Methods for Ordinary Differential Systems (Wiley, New York, 1991).
V. X. Liu, Instability for the Navier-Stokes equations on the 2-dimensional torus and a lower bound for the Hausdorff dimension of their global attractors.Commun. Math. Phys. 147:217–230 (1992); A sharp lower bound for the Hausdorff dimension of the global attractors 2D Navier-Stokes equations,Commun. Math. Phys. 158:327–339 (1993).
V. X. Liu, On unstable and neutral spectrum of incompressible inviscid and viscid fluids on the 2D torus,Q. Appl. Math. (1996), to appear.
C. Marchioro, An example of absence of turbulence for any Reynolds number.Commun. Math. Phys. 105:99–106 (1986).
J. E. Marsden and M. McCracken,The Hopf Bifurcation and Its Applications (Springer, Berlin, 1976).
M. S. Massey,A Basic Course in Algebraic Topology (Springer, Berlin, 1991).
L. D. Meshalkin and Ya. G. Sinai, Investigation of the stability of a stationary solution of a system of equations for the plane movement of an incompressible viscous fluid,J. Math. Mech. 19(9):1700–1705 (1961).
H. Okamoto and M. Shoji, Bifurcation diagrams in Kolmogorov's problem of viscous incompressible fluid on 2-D flat tori,Kyoto Univ. Res. Inst. Math. Sci. 767:29–51 (1991).
R. Temam,Navier-Stokes Equations (North-Holland, Amsterdam, 1983).
E. S. Titi, On a criterion for locating stable stationary solutions to the Navier-Stokes equations,Nonlinear Anal. TMA 11:1085–1102 (1987).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Chen, ZM., Price, W.G. Long-time behavior of Navier-Stokes flow on a two-dimensional torus excited by an external sinusoidal force. J Stat Phys 86, 301–335 (1997). https://doi.org/10.1007/BF02180208
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF02180208