TY - JOUR

T1 - Hard turbulence in a finite dimensional dynamical system?

AU - Bartuccelli, Michele

AU - Constantin, Peter

AU - Doering, Charles R.

AU - Gibbon, John D.

AU - Gisselfält, Magnus

N1 - Funding Information:
We thank J.M. Hyman, C.D. Levermore, D. McLaughlin and H. Rose for helpful discussions. We are especially grateful to D.D. Holm for extensive discussions and encouragement, and for hosting three of us (PC, CRD and JDG) at Los Alamos National Laboratory where part of this work was completed. Financial support of this research came in part from NSF grants DMS-8 602031, PHY-8958506 and PHY-8907755. MB thanks the UK SERC for the award of a Research Assistantship. PC is an Alfred P. Sloan Research Fellow, and CRD is a Presidential Young Investigator.

PY - 1989/12/18

Y1 - 1989/12/18

N2 - 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.

AB - 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.

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U2 - 10.1016/0375-9601(89)90380-0

DO - 10.1016/0375-9601(89)90380-0

M3 - Article

AN - SCOPUS:5544322654

SN - 0375-9601

VL - 142

SP - 349

EP - 356

JO - Physics Letters A

JF - Physics Letters A

IS - 6-7

ER -