Optimal estimate of the spectral gap for the degenerate Goldstein-Taylor model

Etienne Bernard (1) , Francesco Salvarani (2)

1 IGN - Institut National de l'Information Géographique et Forestière [IGN]
2 Dipartimento di matematica F. Casorati

In this paper we study the decay to the equilibrium state for the solution of a generalized version of the Goldstein-Taylor system, posed in the one-dimensional torus $\T=\R/\Z$, by allowing that the non-negative cross section $\sigma$ can vanish in a subregion $X:=\{ x \in \T\, \vert \, \sigma(x)=0\}$ of the domain with $\text{meas}\,(X)\geq 0$ with respect to the Lebesgue measure. We prove that the solution converges in time, with espect to the strong $L^2$-topology, to its unique equilibrium with an exponential rate whenever $\text{meas}\,(\T \setminus X)\geq 0$ and we give an optimal estimate of the spectral gap.

Domaines: Equations aux dérivées partielles [math.AP]

Etienne Bernard, Francesco Salvarani. Optimal estimate of the spectral gap for the degenerate Goldstein-Taylor model. Journal of Statistical Physics, 2013, 153 (2), pp.363-375. ⟨10.1007/s10955-013-0825-6⟩. ⟨hal-00766211v2⟩