Abstract
We consider the Goldstein-Taylor model, which is a 2-velocity BGK model, and construct the “optimal” Lyapunov functional to quantify the convergence to the unique normalized steady state. The Lyapunov functional is optimal in the sense that it yields decay estimates in \(L^2\)-norm with the sharp exponential decay rate and minimal multiplicative 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 optimal decay rates of 2-dimensional ODE systems, and a partial answer for higher dimensional ODE systems.
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Notes
- 1.
An eigenvalue is defective if its geometric multiplicity is strictly less than its algebraic multiplicity.
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Acknowledgments
All authors were supported by the FWF-funded SFB #F65. The second author was partially supported by the FWF-doctoral school W1245 “Dissipation and dispersion in nonlinear partial differential equations”. We are grateful to the anonymous referee who led us to better distinguish the different cases studied in §3 and §4.
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Achleitner, F., Arnold, A., Signorello, B. (2019). On Optimal Decay Estimates for ODEs and PDEs with Modal Decomposition. In: Giacomin, G., Olla, S., Saada, E., Spohn, H., Stoltz, G. (eds) Stochastic Dynamics Out of Equilibrium. IHPStochDyn 2017. Springer Proceedings in Mathematics & Statistics, vol 282. Springer, Cham. https://doi.org/10.1007/978-3-030-15096-9_6
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