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Bi-Lipschitz Mané projectors and finite-dimensional reduction for complex Ginzburg-Landau equation
1University of Surrey, Department of Mathematics, Guildford, Surrey GU2 7XH, UK.
Proceedings. Mathematical, Physical, and Engineering Sciences
|August 25, 2020
Summary
We developed a novel method to prove finite-dimensionality for complex systems with cross-diffusion. This approach uses spatial and temporal averaging to analyze limit dynamics, applicable to models like the Ginzburg-Landau equation.
Area of Science:
- Dynamical Systems
- Partial Differential Equations
- Mathematical Physics
Background:
- Semilinear parabolic systems with cross-diffusion terms exhibit complex dynamics.
- Establishing finite-dimensionality of these dynamics is crucial for analysis.
- The complex Ginzburg-Landau equation is a relevant model system.
Purpose of the Study:
- To present a new method for proving finite-dimensionality of limit dynamics.
- To apply this method to a three-dimensional complex Ginzburg-Landau equation.
- To analyze the impact of cross-diffusion terms on system dynamics.
Main Methods:
- Combining the spatial-averaging principle (Sell and Mallet-Paret) with temporal averaging.
- Utilizing bi-Lipschitz Mané projectors to characterize limit dynamics.
- Analyzing systems with cross-diffusion terms.
Main Results:
- Successfully established the finite-dimensionality of limit dynamics for the model system.
- Demonstrated the effectiveness of the combined averaging technique.
- Provided insights into the behavior of solutions influenced by cross-diffusion.
Conclusions:
- The developed method offers a robust framework for analyzing complex parabolic systems.
- Finite-dimensionality is confirmed for the Ginzburg-Landau model with cross-diffusion.
- The findings contribute to the understanding of global attractors in dissipative systems.
Keywords:
Lipschitz Mané projectorscomplex Ginzburg–Landau equationinertial manifoldslarge dispersionspatial averaging principletemporal averagingMore Related Videos
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