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.Key words and phrases. Lyapunov functionals, sharp decay estimates, Goldstein-Taylor model.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.

The Goldstein-Taylor equations can be thought of as a simplified version of a BGK system, where the velocity variable is constricted to a discrete set of values. It is intimately related to turbulent fluid motion and the telegrapher’s equation. A detailed understanding of the large time behaviour of the solutions to these equations has been mostly achieved in the case where the relaxation function, measuring the intensity of the relaxation towards equally distributed velocity densities, is constant. The goal of the presented work is to provide a general method to tackle the question of convergence to equilibrium when the relaxation function is not constant, and to do so as quantitatively as possible. In contrast to the usual modal decomposition of the equations, which is natural when the relaxation function is constant, we define a new Lyapunov functional of pseudodifferential nature, one that is motivated by the modal analysis in the constant case, that is able to deal with full spatial dependency of the relaxation function. The approach we develop is robust enough that one can apply it to multi-velocity Goldstein-Taylor models, and achieve explicit rates of convergence. The convergence rate we find, however, is not optimal, as we show by comparing our result to those found in [8].

We are concerned with the short-and large-time behavior of the L 2 -propagator norm of Fokker-Planck equations with linear drift, i.e. ∂tf = divx(D∇xf + Cxf ). With a coordinate transformation these equations can be normalized such that the diffusion and drift matrices are linked as D = C S , the symmetric part of C. The main result of this paper (Theorem 3.1) is the connection between normalized Fokker-Planck equations and their drift-ODE ẋ = −Cx: Their L 2 -propagator norms actually coincide. This implies that optimal decay estimates on the drift-ODE (w.r.t. both the maximum exponential decay rate and the minimum multiplicative constant) carry over to sharp exponential decay estimates of the Fokker-Planck solution towards the steady state. A second application of the theorem regards the short-time behaviour of the solution: The short-time regularization (in some weighted Sobolev space) is determined by its hypocoercivity index, which has recently been introduced for Fokker-Planck equations and ODEs (see [F.

We are concerned with the short-and large-time behavior of the L 2 -propagator norm of Fokker-Planck equations with linear drift, i.e. ∂ t f = div x (D∇ x f + C x f ). With a coordinate transformation these equations can be normalized such that the diffusion and drift matrices are linked as D = C S , the symmetric part of C . The main result of this paper is the connection between normalized Fokker-Planck equations and their drift-ODE ẋ = −C x: Their L 2 -propagator norms actually coincide. This implies that optimal decay estimates on the drift-ODE (w.r.t. both the maximum exponential decay rate and the minimum multiplicative constant) carry over to sharp exponential decay estimates of the Fokker-Planck solution towards the steady state. A second application of the theorem regards the short time behaviour of the solution: The short time regularization (in some weighted Sobolev space) is determined by its hypocoercivity index, which has recently been introduced for Fokker-Planck equations and ODEs (see [5,1,2]). In the proof we realize that the evolution in each invariant spectral subspace can be represented as an explicitly given, tensored version of the corresponding drift-ODE. In fact, the Fokker-Planck equation can even be considered as the second quantization of ẋ = −C x.

This paper is concerned with finding Fokker-Planck equations in R d with the fastest exponential decay towards a given equilibrium. For a prescribed, anisotropic Gaussian we determine a non-symmetric Fokker-Planck equation with linear drift that shows the highest exponential decay rate for the convergence of its solutions towards equilibrium. At the same time it has to allow for a decay estimate with a multiplicative constant arbitrarily close to its infimum.Such an "optimal" Fokker-Planck equation is constructed explicitly with a diffusion matrix of rank one, hence being hypocoercive. In an L 2 -analysis, we find that the maximum decay rate equals the maximum eigenvalue of the inverse covariance matrix, and that the infimum of the attainable multiplicative constant is 1, corresponding to the high-rotational limit in the Fokker-Planck drift.