Abstract
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].
Highlights
The object of this work is the large time analysis of the Goldstein-Taylor equations on the onedimensional torus T, i.e. on [0, 2π] with periodic boundary conditions, and for t ∈ (0, ∞):σ (x) ∂t f+(x, t) + ∂x f+(x, t) = 2 ( f−(x, t) − f+(x, t)),∂t f−(x, t) − ∂x f−(x, t) = − σ (x 2 ) ( f−(x, t f+(x, t)), (1)
As the main driving force of the equation is a transport operation on the torus, we will not be surprised to learn that the large time behaviour of u is convergence to a constant
Remark 1 It is simple to see that if σ (x) satisfies the conditions of (b), as σmin and σmax approach a positive constant σ = 2, we find that
Summary
The object of this work is the large time analysis of the Goldstein-Taylor equations on the onedimensional torus T, i.e. on [0, 2π] with periodic boundary conditions, and for t ∈ (0, ∞):. The Goldstein-Taylor model was originally considered as a diffusion process, resulting as a limit of a discontinuous random migration in 1D, where particles may change direction with rate σ It appeared in the context of turbulent fluid motion and the telegrapher’s equation, see [15,22], respectively. The precise value of this spectral gap, is hardly accessible - even for simple non-constant relaxation functions σ (x) (see e.g. Appendix A) It is based on the restrictive requirement f±,0 ∈ H 1(T), and cannot be extended to other discrete velocity models in 1D. In Appendix A we discuss a potential way to improve the technique from §5, and explicitly show the lack of optimality of it for a particular case of σ (x)
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