April  2001, 7(2): 403-429. doi: 10.3934/dcds.2001.7.403

The Lorenz equation as a metaphor for the Navier-Stokes equations

1. 

Department of Mathematics, Indiana University,Bloomington, IN, 47405, United States, United States

2. 

Department of Mathematics,, The University of Southern California, Los Angeles, CA 90089-1113, United States

3. 

Department of Mathematics and Department of Mechanics and Aerospace Engineering, University of California, Irvine, CA 92697, United States

Revised  November 2000 Published  January 2001

Three approaches for the rigorous study of the 2D Navier-Stokes equations (NSE) are applied to the Lorenz system. Analysis of time averaged solutions leads to a description of invariant probability measures on the Lorenz attractor which is much more complete than what is known for the NSE. As is the case for the NSE, solutions on the Lorenz attractor are analytic in a strip about the real time axis. Rigorous estimates are combined with numerical computation of Taylor coefficients to estimate the width of this strip. Approximate inertial forms originally developed for the NSE are analyzed for the Lorenz system, and the dynamics for the latter are completely described.
Citation: C. Foias, M. S Jolly, I. Kukavica, E. S. Titi. The Lorenz equation as a metaphor for the Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2001, 7 (2) : 403-429. doi: 10.3934/dcds.2001.7.403
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