F. Achleitner, A. Arnold, B. Signorello:

"On optimal decay estimates for ODEs and PDEs with modal decomposition";

in: "ASC Report 4/2018", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2018, ISBN: 978-3-902627-11-7, 1 - 14.

We consider the Goldstein-Taylor model, which is a 2-velocity

BGK model, and construct the "optimal" Lyapunov functional to quan-tify the convergence to the unique normalized steady state. The Lya-punov functional is optimal in the sense that it yields decay estimates in L2-norm with the sharp exponential decay rate and minimal multiplica-tive constant. The modal decomposition of the Goldstein-Taylor model leads to the study of a family of 2-dimensional ODE systems. Therefore we discuss the characterization of "optimal" Lyapunov functionals for linear ODE systems with positive stable diagonalizable matrices. We give a complete answer for 2-dimensional ODE systems, and a partial answer for higher dimensional ODE systems.

Lyapunov functionals, sharp decay estimates, Goldstein-Taylor model.

http://www.asc.tuwien.ac.at/preprint/2018/asc04x2018.pdf

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