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Optimal Estimate of the Spectral Gap for the Degenerate Goldstein-Taylor Model

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Abstract

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 \({\mathbb{T}}={\mathbb{R}}/{\mathbb{Z}}\), by allowing that the nonnegative cross section σ can vanish in a subregion \(X:=\{ x \in {\mathbb{T}}\, \vert\, \sigma(x)=0\}\) of the domain with meas (X)≥0 with respect to the Lebesgue measure.

We prove that the solution converges in time, with respect to the strong L 2-topology, to its unique equilibrium with an exponential rate whenever \(\text{meas}\,({\mathbb{T}}\setminus X)\geq0\) and we give an optimal estimate of the spectral gap.

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  • 14 September 2020

    In [1], on p. 366, line 5 and on p. 369, line 12, replace

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Acknowledgements

The authors are grateful to the referees for their useful suggestions and comments concerning the paper. This paper has been partially supported by the Italian national institute of higher mathematics (INDAM), GNFM project “Study of complex kinetic systems: theoretical analysis and numerical simulation”.

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Authors and Affiliations

  1. Institut Géographique National, Laboratoire de Recherche en Géodésie, Université Paris Diderot, Bâtiment Lamarck A, 5, rue Thomas Mann, Case courrier 7071, 75205, Paris Cedex 13, France

    Étienne Bernard

  2. Dipartimento di Matematica F. Casorati, Università degli Studi di Pavia, Via Ferrata 1, 27100, Pavia, Italy

    Francesco Salvarani

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  1. Étienne Bernard
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  2. Francesco Salvarani
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Correspondence to Francesco Salvarani.

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Bernard, É., Salvarani, F. Optimal Estimate of the Spectral Gap for the Degenerate Goldstein-Taylor Model. J Stat Phys 153, 363–375 (2013). https://doi.org/10.1007/s10955-013-0825-6

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