Hard turbulence in a finite dimensional dynamical system?

We present new time-averaged and time-asymptotic bounds on various norms of solutions to the 2-d complex Ginzburg-Landau (CGL) equation, ∂A ∂t = RA + (1 + iv)ΔA - (1 + iμ)|A|²A. These bounds establish the existence of a finite dimensional global attractor and inertial manifolds, so that the dynamics on the attractor are those of a finite dimensional dynamical system. The CGL equation is known to have chaotic solutions characterized by relatively long length-scale, low-modal dynamics, also known as soft turbulence. Our new estimates suggest that near the nonlinear Schrödinger (NLS) limit (|v|, |μ| →∞, if and only it vμ<0), solutions of the CGL equation are dominated by the remnants of the blow-up solutions of the NLS equation, resulting in asymptotic dynamics marked by large intermittent spikes in space and time, i.e., hard turbulence.