On the behavior of the Lorenz equation backward in time
Studying two-dimensional periodic Navier-Stokes equations, \textit{P. Constantin, C. Foias, I. Kukavica} and \textit{A. J. Majda} [J. Math. Pures Appl. (9) 76, 125--153 (1997; Zbl 0874.35085)] established existence of rich invariant sets formed by trajectories with given exponential growth backward in time. Each solution with backward exponential growth has an eigenvalue of the Stokes operator as its exponential rate. Furthermore, the set \(M_{\Lambda}\) of solutions that exist for all times and have exponential rate not exceeding \(\Lambda\) projects onto the linear space spanned by all eigenvectors of the Stokes operator corresponding to eigenvalues not exceeding \(\Lambda.\) It has been noted that the Lorenz system can serve as a low-dimensional paradigm for the two-dimensional periodic Navier-Stokes equations. \textit{C. Foias, M. S. Jolly, I. Kukavica} and \textit{E. S. Titi} [Discrete Con. Dyn. Syst. 7, 403--429 (2001; Zbl 1052.76012)] explored some global properties of the Lorenz system which are analogous to those of the Navier-Stokes equations. In particular, three invariant sets for the Lorenz system, \(M_{1},\) \(M_{b},\) and \(M_{\sigma}\) were introduced and examined. In this paper, by adapting to the Lorenz system the methods exploited by Constantin et al., op. cit., the authors establish that the invariant set \(M_{b}\) has Hausdorff dimension not less than 2 and is contained in the exterior of a cone, that is, the two sides of the cone are in \(M_{\sigma}\setminus M_{b}.\) Furthermore, \(M_{b}\) projects onto the \(xy\)-plane, which perfectly agrees with one of the key results established by Constantin et al., op. cit. Several interesting motivations for the study of the backward behavior of dissipative evolutionary equations (including the Lorenz system) are provided.
backward behavior, Navier-Stokes equations
scientific article; zbMATH DE number 2136540
scientific article